(written from a Production point of view)
Universal Studios, aka Universal Pictures or Universal for short, is one of the subsidiaries of NBCUniversal and, having been founded on 30 April 1912, the oldest motion picture studio in Hollywood. The second oldest, though only by eight days, is Paramount Pictures, which was founded on 8 May 1912.
Star Trek connections
While owned and produced by Paramount Pictures, and later by CBS Corporation, Universal has made contributions to the Star Trek franchise, but these were usually not directly discernible to the public eye.
Being industry competitors notwithstanding, Paramount and Universal joined forces when they, as equal partners, established Cinema International Corporation (CIC) in 1971 (as of 1981: United International Pictures – UIP – ) as a joint venture, and responsible for the distribution of feature films outside the US, which included all of the later Star Trek films. This action was necessitated partly for cost-cutting reasons, partly for the 1948 United States Supreme Court anti-trust ruling leveled against the industry – most ironically against Paramount Pictures in particular, they being singled out by the Court to serve as proxy for the entire industry – , specifically aimed to break the hold individual studios hitherto had on the entirety of the industry, otherwise known as the traditional "Hollywood Studio System" (see also in this regard: Desilu Studios and National Amusements). With the advent of the VHS and Betamax home video tape format, a subsidiary division, CIC Video, was established two years later, responsible for the distribution of this home video format – exclusively through the rental circuit initially – , including all the Star Trek productions released in this format. Yet, CIC Video as a joint venture was dissolved in 1999 (corresponding with the demise of the video tape in favor of such later home video formats as the LaserDisc, VCD, DVD, and later still, the Blu-ray Disc) when Paramount reasserted full control over the release of their home video formats through their own division, Paramount Home Entertainment. As of 2016 however, UIP is still in operation and still equally shared by Paramount and Universal.
A notable 50/50 joint venture became USA Networks, which in 1992 picked up the original Sci-Fi Channel proposition, a national cable and satellite television channel. The initial advisory board counted among its members Star Trek creator Gene Roddenberry and his friend Isaac Asimov, but both had already died before the channel was launched on 24 September 1992.  The network aired Star Trek: The Original Series and Star Trek: The Animated Series, until the partnership sold off the network in 1997. However in 2002, Universal rebought the channel as sole owner, in anticipation of the new Battlestar Galactica television series project they had in development. In 2007, the as SyFy renamed network acquired the rights to the then most recent Star Trek series, Star Trek: Enterprise, which began airing on 8 January 2007. By then Universal had merged with longstanding industry partner NBC, the original broadcaster of The Original and Animated Series, into NBCUniversal as of 2004.
Star Trek Adventure
It was not only for the distribution of Star Trek live-action productions, that the two industry competitors cooperated; while licensed by Paramount, it was somewhat ironic that it was not the official franchise, but rather Universal who produced and organized one of the very first official Star Trek attractions, the two 1988-1996 variant, largely overlapping "Star Trek Adventure" live-performance attractions, housed on its two "Universal Studios Tour" theme-park premises in Hollywood, Los Angeles, California and in Orlando, Florida. 
Sharing production assets
On later occasions, parts of the episodes "The Killing Game", "The Killing Game, Part II", "Fair Haven", and "Spirit Folk" were filmed on location at Universal Studios in California on a redress of their "European Street" backlot set, as the exteriors of the cities Sainte Claire and Fair Haven. Some of the buildings can be seen in both episodes. For the Star Trek: Enterprise episode "North Star", the production crew went to the Universal Studios backlot set "Western Town", which was used to portray the city. The Star Trek: Picard first season episodes "Absolute Candor" and "Nepenthe" also used the Universal backlot as a filming location. The former episode filmed the Vashti colony scenes at the Mexican Street backlot, while the latter utilized a forest log cabin built for the 1988 film The Great Outdoors as the Troi-Riker house. "Stardust City Rag" featured location work at Hollywood CityWalk, next to Universal Studios.
The friendly cooperation between the two studios was, aside from the use of each other's standing sets, also exemplified by Universal allowing the Star Trek production to occasionally use production materials and clippings from their (film) productions as recycled footage. Two such examples were the re-use of the Relva VII outpost matte painting for TNG: "Coming of Age", and the use of clippings from Universal's 2000 war film U-571 seen in the title sequences of the two mirror universe episodes, ENT: "In a Mirror, Darkly".
Despite the cooperation with Paramount, Universal was one of the Hollywood studios that became increasingly envious of Paramount for its Star Trek franchise due to the property's stable and highly profitable revenue stream for that studio, especially in the early-to-mid 1990s when Star Trek was at its peak in popularity, being Paramount's most profitable property for a period of time, admitted as such by them. (A Vision of the Future - Star Trek: Voyager, pp. 50-51) Having already missed out on the Star Wars franchise when it turned down its creator, George Lucas, in the mid-1970s , Universal became one of the studios that invested most heavily trying to create something akin to the Star Trek franchise for themselves – the other ones being Warner Brothers (Babylon 5, 1993-2002) and 20th Century Fox (Space: Above and Beyond, 1995-1996; Firefly, 2002-2005; Terra Nova, 2011) which, though having been the production company, declined to become further involved in Star Wars as a franchise.
Early attempts by Universal to create a science fiction franchise of their own, at that time prompted on by the phenomenal success of the first 1977 Star Wars movie and on which they had missed out on, encompassed Glen A. Larson's original Battlestar Galactica franchise (1978-1979) and his follow-up Buck Rogers in the 25th Century (1979-1981, and both productions served by future Star Trek production staffers), but both did not succeed in achieving the hoped for success, neither surpassing a two-season run, even though both have attained something of a cult status later on. It was not only Universal Studios that was inspired by Star Wars, Paramount too was inspired by the movie to such a degree that they made the ad hoc decision to turn the Star Trek: Phase II television series they had under development at the time into a major theatrial motion picture, eventually even bringing aboard the same visual effects team, that Universal had used on its Galactica pilot, John Dykstra's Apogee, Inc. and who himself had been instrumental for the Star Wars visual effects, while employed by Industrial Light & Magic. (see: The Motion Picture: Production history)
Universal's first serious (post-Star Trek: The Next Generation) emulation attempt, seaQuest DSV (1993-1996), was an unadulterated expensive flop. Future Star Trek: Voyager, but then seaQuest, production staffer Ben Betts elaborated, "They definitely wanted to have something like Star Trek. They wouldn't say that aloud, but that was what they were going for. They were trying to find Star Trek under water. Everything was there, except for the stories. They didn't have enough of a human element so they'd get caught up in the technology...kind of fall back on the technology to bail everybody out by the end of the episode. It was plain as day to people working on the show. Everything was right. They were spending the money to make the graphics look good, the CGI [note: produced by Amblin Imaging, especially established for SeaQuest, and later working on the CGI of the first two seasons of Voyager] looked great, the sets were well lit, they had a pretty good cast...but it didn't work. It still wasn't Star Trek. (A Vision of the Future - Star Trek: Voyager, p. 54) Becoming a minor, obscure footnote in science fiction television history, seaQuest DSV has all but been forgotten.
Far more successful was Universal's second attempt, when they revisited their own older, nearly forgotten and short-lived science fiction property: Ronald D. Moore's re-imagined 2003-2012 Battlestar Galactica franchise, the new series they had in development in 2002, which started airing on Universal's recently (re-)acquired SyFy Channel. Though highly successful in the time period 2003-2009, both in popularity and acclaim – especially in light of the, at the time, very poorly received Star Trek: Enterprise – , while the primary revamped Galactica series was in production, after production was suspended, interest in Battlestar Galactica as a franchise petered out quickly, particularly due to the failure of its spin-off productions (most notably the Caprica series and therefore sharing a similar fate with Warner Brothers' Babylon 5 franchise), and it had nowhere near the longevity the Star Trek franchise had enjoyed. Unlike seaQuest DSV though, Battlestar Galactica, both the revamped version as well as its original 1970s progenitor, does enjoy a cult status.
Universal's third attempt however, the more recent Defiance series (2013-2015, and on which – like the revamped Battlestar Galactica – many former Star Trek visual effects staffers had worked), has fared little better than seaQuest DSV.
Incidentally, the Hollywood studio that came closest to emulate the longevity and format of the Star Trek franchise, had been MGM, ironically the very first studio to turn down Star Trek: The Original Series back in 1964, with its Stargate franchise (its primary series Stargate SG-1 ran for an all-time science fiction show record breaking ten seasons), which ran from 1994 through 2011, though interest in that franchise, despite its likewise cult status, has waned too, after live-action production had been suspended in 2011.