(written from a Production point of view)
The Simpsons is an animated cartoon show created by Matt Groening, also the creator of Futurama, which debuted on December 17, 1989, making it one of the longest running shows in American history. Over the course of the show's run, The Simpsons has referenced Star Trek numerous times.
Frank Welker has done sounds of various animals in the show including the family dog Santa's Little Helper.
Kang and Kodos
Perhaps the most common Star Trek reference on the show is the naming of the two recurring alien villains. Typically appearing in "Treehouse of Horror" episodes, Kang and Kodos also claim to be native to the planet Rigel VII.
"Bart of Darkness"
The Itchy and Scratchy cartoon in this episode is a parody of TOS: "The Cage", "The Menagerie, Part I", and "The Menagerie, Part II", with Itchy resembling the aliens from Talos IV and possessing strong telepathic and telekenetic abilities.
"Itchy and Scratchy: The Movie"
The family sees a trailer and clips from the latest Original Series film, Star Trek XII: So Very Tired, which depicts an elderly and senile cast. Captain Kirk complains about poor sleeping habits and everyone else's indifference to his complaining, while Scotty has grown so fat he cannot fire the phasers because his gargantuan stomach is in the way of the control panel.
"Marge vs. the Monorail"
Leonard Nimoy guest-stars as himself where he makes a celebrity apperance at the christening of the Springfiled monorail, where he says it could do at least Warp 5. At the end of the episode, he makes his exit by transporting. He also mentions that the doors in Star Trek: The Original Series were not automated, but operated by stagehands, which is, in fact, true. At the end of the episode, Nimoy states "My work is done here." and beams out of the scene.
"Deep Space Homer"
The episode's title is a reference to Star Trek: Deep Space Nine. As part of his NASA training, Homer Simpson fights Barney Gumble in the Triskelion arena with the classic "Star Trek fight music" in the background while one of the NASA administrators bets "quatloos on the newcomer." Later, Homer, Bart, and Lisa watch an episode of Itchy and Scratchy entitled "Scar Trek: The Next Laceration", which has otherwise no relation to Star Trek except for its taking place in space.
In this episode, Hugh's watch makes identical noises as a TOS communicator, Troy McClure's uniform is identical to the TOS uniforms, and the nuclear power plant resembles and makes the same noises as the Enterprise.
"The Springfield Files"
At one point, Dr. Hibbert asks Homer if the alien he saw is carbon-based or silicon-based, a reference to the Horta.
"Two Dozen and One Greyhounds"
A clerk at a pet store performs a "Human-dog mind meld" on Santa's Little Helper. He recommends that the Simpsons buy a lot of his merchandise to help calm the dog – apparently it worked as the next scene depicts the family driving home with a car full of dog related goods.
"Mayored to the Mob"
The Simpsons go to a sci-fi convention, featuring Star Trek references, including uniforms, etc. Also, the convention's welcome motto is "Set your phasers on fun." Vina's theme from "The Cage" is also heard when Comic Book Guy is entranced by a nerdy young Comic Book Gal.
When Marge must replace all of the family's photos that were lost in a photo album fire, she recreates the family's "Star Trek: Voyager Series Finale Party" with Lenny dressed as Seven of Nine and Dr. Hibbert dressed as Tuvok. Carl begins to cry as he thought he could make it through Voyager's final episode again, but realizes he can't, while Homer wails, "Oh, Captain Janeway! Your mission ended too soon! Too soon!"
"Tis the Fifteenth Season"
Bart shows Homer different TV versions of A Christmas Carol. One of them is a TOS version where the Ghost of Christmas Future appears to the crew as a light being. Kirk orders Scotty to fire the torpedoes, but he cannot because the ghost has shown him a vision of how fat he is in the future.
"My Big Fat Geek Wedding"
Comic Book Guy and Edna Krabappel participate in a Klingon wedding ceremony at the "bi-monthly science fiction convention" (Bi-Mon-Sci-Fi-Con). Later Homer and Marge are remarried by the Klingon priest.
"Stealing First Base"
Bart accidentally falls off the school roof and a girl that he had a crush on gives him CPR. As she starts, a montage of different kisses from movies are shown. One of the kisses featured was from J.J. Abrams' Star Trek where Uhura kissed Spock.
"The Squirt and The Whale"
After Lisa tries to save a beached whale from dying, but failed, the citizens of Springfield make use of its remains. One use was for corsets, which Comic Book Guy wears along with Kirk's TOS-uniform. He then proclaims that he is Captain Kirk from Star Trek I, but then the corset gives way little by little and he corrects himself from I to II to V to Generations, until the corset finally gives out and Comic Book Guy says "Boston Legal".
"The Day The Earth Stood Cool"
Also, when Bart is fighting T-Rex, the band in the factory starts playing the Star Trek fight music.
Homer aims to live out his unfulfilled dreams with the help of his childhood penpal, including staging a recreation of "Arena". They rent costumes from Comic Book Guy for only $10, though he requests they not stink up the Gorn head as he needs it for a Bar Mitzvah later. The Gorn costume is miscolored purple for the first of the two shots it appears in.
"The Man Who Came to Be Dinner"
This episode concludes with a series of Star Trek homages including end credits spoofing that of the original series. The Simpsons fly a starship in a bridge based on the Enterprise's, complete with original sound effects. (Homer is the Captain; Marge is the commications officer; Bart is Chief enginner; Lisa is Science officer and Maggie is the navigator/helmsman) On the viewscreen, they mistake Grandpa for a Klingon, with a leitmotif based on the one composed by Jerry Goldsmith. The end credits proceed to depict various Simpsons characters in classic Star Trek episodes.  (Lenny and Carl in "Let That Be Your Last Battlefield"; Barney and Patti and Selma in "I, Mudd": Homer reenacting "Arena"; Crazy Cat Woman and cats in "The Trouble with Tribbles"; Bart Simpson as Engineer Scott in a Jefferies Tube "The Doomsday Machine"; Marge Simpson as Vina in her Orion slave girl guise in "The Cage"; Milhouse as Quark; Mr Burns as the Balok Puppet in "The Corbomite Maneuver".)
"Lisa with an 'S'"
Other references and minor parodies
- In "Homer vs. Patty and Selma" and "Itchy and Scratchy Land", the couch gag shows the family transporting onto the couch using transporters seen in TNG.
- In "Blood Feud", Mr. Burns uses the phrase "Revenge is a dish best served cold." from Star Trek II: The Wrath of Khan.
- In "A Star is Burns", William Shatner is one of the actors auditioning for the role of Montgomery Burns dressed in a Starfleet uniform and speaking in his "Kirk voice" ("Ex-cell-ENT!").
- In "The Homer They Fall", the Comic Book Guy tries to sell an utility belt he won in a bingo game at a Star Trek convention to a pawn shop. He explains that the organizers have severely miscalculated, as a medium-sized belt is obviously too small for most Star Trek fans. The shopkeeper refuses to buy the belt, and eventually Bart buys it for four dollars.
- In "Das Bus", Comic Book Guy attempts to download a nude photo of Captain Janeway only to be interrupted by an ad for Homer's internet company; Comic Book Guy speculates if the service could provide "faster nudity."
- In "Treehouse of Horror IX", After Lisa puts a radioactive battery in the remote control, she presses the button of the remote to the TV, the remote fires at the TV with the same sound as the Federation hand phasers used in Star Trek: The Next Generation, Star Trek: Deep Space Nine, Star Trek: Voyager, Star Trek Generations, and Star Trek: First Contact.
- In "Treehouse of Horror X", one of Comic Book Guy's (as arch-nemesis "The Collector") plastic-wrapped acquisitions in his lair is Jeri Ryan in her costume and prosthetic appliances as Seven of Nine and holds Bart and Lisa at bay with "the only working phaser ever built" which was "fired once to keep William Shatner from releasing another album".
- In "Bart's Girlfriend", Grandpa Simpson admits that he was "the one that canceled Star Trek".
- In "Old Yeller Belly", the Comic Book Guy can be seen taking back a Spider-Man and a Batman outfit, and an Original Series blue sciences uniform to a costume store.
- In "Simpson Tide", Homer ends up as captain of a military submarine and refers to helmsman Moe Szyslak as "Mr. Moe". A "Mr. Sulu" is also part of the crew and mentions Rigel VII while setting a course.
- In "I'm Goin' to Praiseland", the Comic Book Guy hallucinates he is Spock, and saves Kirk from a haywire chair. Uhura declares she wants to make out with him, as do Catwoman and Agent 99.
- In "Homer vs. Dignity", Homer tells Comic Book Guy he'd like to buy a mint-condition Spider-Man issue #1, who retorts "and I'd like an hour on the holodeck with Seven of Nine!"
- In "Sweets and Sour Marge", the Comic Book Guy is seen buying Leonard Nimoy's three books, I Am Not Spock, I Am Spock and I Am Also Scotty (of course, the third being only a fictional book).
- In "The Father, the Son and the Holy Guest Star", Finnegan's theme from TOS: "Shore Leave" (composed by Gerald Fried) can be heard during an Irish brawl in "Catholic Heaven".
- In "Mypods and Broomsticks" the Mapple headquarters door makes the same sound of the door sliding open in TOS.
- The story "Fan-Tasty Island" (From Simpsons Comics #10) starts off with Homer dreaming that he is a captain in Starfleet, with his crew consisting of Apu Nahasapeemapetilon, Lenny Leonard and Groundskeeper Wille/chief engineering officer (all of whom are wearing the Starfleet uniform from late 2270s-2350s). Before waking up, Homer takes an interest in a giant donut surrounding an unnamed planet.
- In addition to Leonard Nimoy, other actors/actresses have provided voice work for the show. Those voices include Wren T. Brown, Kim Cattrall, Christopher Collins, Jim Cummings, Andy Dick, Miriam Flynn, Kelsey Grammer, Jess Harnell, John Kassir, Maurice LaMarche, Tress MacNeille, Colm Meaney, Charles Napier, Winona Ryder, Jack Sheldon, Sarah Silverman, Patrick Stewart, George Takei, Ron Taylor, Scott Thompson, Karl Wiedergott, Paul Willson, Paul Winfield, Brent Spiner and Seth MacFarlane.