By 2256, one Beatles cover band included Paul Stamets' uncle Everett. That year, Stamets pointed out to Michael Burnham the obvious fact that, even though Everett performed in a Beatles cover band, it didn't mean he was John Lennon. (DIS: "Context Is for Kings")
In canon, this band has been mentioned only by the common shortening "Beatles".
In reality, The Beatles, which were led by primary songwriters John Lennon and Paul McCartney, have been referenced multiple times in behind-the-scenes documentation about Star Trek. Marc Cushman, writer of several Star Trek reference books, likened Star Trek: The Original Series to The Beatles, stating, "One thing about Star Trek that I've said before, and I really believe it, is it was The Beatles of 1960s TV. And if you had to describe The Beatles, you would say it's magic, and take any one of them out of that band and it's not The Beatles. Well, Star Trek's the same way, from the same period. I mean, take William Shatner out, take Leonard Nimoy out, take [Gene] Roddenberry or [Gene L.] Coon or [D.C.] Fontana out, or DeForest Kelley, and you don't have it. It is still gonna be good, but it's not gonna be what it is." (50 Years of Star Trek) The Simpsons Writer/Producer David Mirkin likewise compared the camaraderie between The Beatles and the TOS cast, remarking that both had "great chemistry." ("Inside The Roddenberry Vault, Part 1", Star Trek: The Original Series - The Roddenberry Vault special features) On the other hand, using an allegory for how the relationship between TOS and TNG might be perceived, TNG writer Tracy Tormé stated, "To some people, The Beatles are just Paul McCartney's band before Wings." (The Fifty-Year Mission: The Next 25 Years, p. 53)
The Beatles and Star Trek became popular at much the same time as each other; as novelist Mark Garland noted, "There was The Beatles, and Star Trek." (Voyages of Imagination, p. 243) Or as Roberto Orci put it, "You know, Star Trek and The Beatles were products of the sixties." (The Fifty-Year Mission: The Next 25 Years, p. 761) The group's connections to the franchise were established very quickly: the February 7, 1964 Ed Sullivan Show episode on which they made their US television debut also featured performances by Charlie Brill, Frank Gorshin and Georgia Brown.
In the final draft script of TOS: "Miri", Jahn was initially described as "perched on an elaborate kid-sized bandstand, complete with the equipment for a Beatles-type trio." Making a bracketed comment seemingly criticizing The Beatles indirectly, the teleplay then stated, "Grandmothers apparently were both indulgent and hard-of-hearing on Earth Two, also." The script additionally characterized Jahn himself as being "dressed as a sort of Beatle, complete with a wig," and pointed out that this was intentional on his part.
The Beatles were also an influence on deciding the look of Pavel Chekov, when he was introduced into Star Trek: The Original Series at the start of its second season. In a memo he sent Casting Director Joseph D'Agosta (dated 22 September 1966), Star Trek creator Gene Roddenberry advised, "Keeping our teenage audience in mind, also keeping aware of current trends, let's watch for a young, irreverent, English-accent 'Beatle' type to try on the show." Later, before casting Walter Koenig as Chekov, Roddenberry ensured that a Beatle-type hairstyle would work on Koenig. (Inside Star Trek: The Real Story, p. 343) Explained Spock actor Leonard Nimoy, "At the same time, The Beatles were a big deal and they had these interesting hair cuts and someone said, 'Maybe we will pick up a bit of The Beatles' audience. We should put a Beatle wig on him.' So Walter came in with his excellent Russian accent and a Beatle wig."  The influence was still acknowledged many decades later: Roberto Orci refers to Chekov as a "little young Beatle" in a DVD bonus feature from the 2009 Star Trek film.
In a letter Gene Roddenberry sent NBC President Herbert Schlosser (dated 1 February 1968), Roddenberry made a comparison between the popularity of Spock and that of The Beatles, noting, "Spock is the hottest item since The Beatles first hit this country."
Marc Cushman likened TOS Season 3 to an altered version of The Beatles, as that season had a reduced budget and its production staff didn't include TOS mainstay writing staffers Gene Roddenberry, D.C. Fontana, Robert Justman, and Gene Coon. "When you take them out of the mix, it's like having The Beatles and taking away John Lennon and Paul McCartney," observed Cushman. "'Okay, we still have George [Harrison] and Ringo [Starr]. We're still The Beatles.' No, you're not. You're still good, but not as good, and that's what you have with the third season." (The Fifty-Year Mission: The First 25 Years)
TNG and DS9 episode writer Morgan Gendel attempted to name the TNG episodes he wrote after songs by The Beatles, as an in-joke. Whereas he was prevented from giving "Starship Mine" the Beatles-inspired title "Revolution" (owing to the existence of the TNG episode title "Evolution"), he finally got his wish with the episode title "The Inner Light", named after a different song by The Beatles. (Captains' Logs: The Unauthorized Complete Trek Voyages, p. 248)
An homage closely related to The Beatles was suggested by Michael Piller and originally written into the anti-time past setting of TNG series finale "All Good Things...". The first draft script of that episode at first involved a character called "Sutcliffe", a USS Enterprise-D conn officer who was upset at the perverseness of Captain Jean-Luc Picard in that era, and finally requested a transfer off the ship. This was an in-joke referring to a "fifth" member of The Beatles – Stuart Sutcliffe, who likewise left his group on the eve of its greatness. (Star Trek: The Next Generation Companion (3rd ed., p. 301))
A musical progression intended to be featured in the film Star Trek was likely to have been by The Beatles; at least according to the script for that film, John Lennon and Paul McCartney wrote a musical progression which Spock had to guess the composers of, while in a skill dome on Vulcan.  In the finished movie, this reference seems at least inaudible among many voices in the scene, or might have been cut entirely. The duo of Lennon and McCartney was also cited as an inspiration for the Kirk-Spock relationship by the film's writers, Alex Kurtzman and Roberto Orci.  According to Orci, this was an allusion to the fact that both Star Trek and The Beatles originated in the 1960s. (The Fifty-Year Mission: The Next 25 Years, p. 761)
AT: "Context Is for Kings" incorrectly stated that "Context Is for Kings" is not the first canonical reference to The Beatles. In the same episode, host Matt Mira speculated that being able to chat with Paul Stamets about The Beatles might be a positive aspect of sharing quarters with Stamets aboard the USS Discovery.
The Alternate Universe series of the Star Trek Customizable Card Game contains an "easter egg" in reference to an infamous urban legend surrounding the group: the Paul Rice card includes the words "daed si luap" (Paul is dead backwards) - a nod to the conspiracy theory that Paul McCartney actually died in 1966, and the Beatles, while replacing him with a lookalike-soundalike, secretly planted messages in their work (often meant to be found by playing songs backwards) which revealed the truth.