Star Trek TV series, films and novels have become a future history and a cultural reference point. When I told a friend about a "reality storm" in Neil Gaiman's The Sandman, he remarked that that sounded like something out of Star Trek, then laughed when I told him that one of the characters had said that that sounded like something out of Star Trek.
A narrator in Robert Heinlein's The Number Of The Beast compares the bridge of Lazarus Long's spaceship to the bridge of the Enterprise - although I would prefer not to refer to The Number Of The Beast.
Isaac Asimov scientifically advised Star Trek. James Blish adapted episodes as short stories and wrote the first Star Trek novel. At a Memorial evening for James Blish in London, Charles Monteith of Faber and Faber described Blish's Cities in Flight future history as "a higher and greater Star Trek."
Larry Niven adapted a Known Space story as a Star Trek animated episode. Niven and Jerry Pournelle place a Chief Engineer from New Scotland on a Navy spaceship and say that this ethnicity is common among Engineers.
If Kirk were in Intelligence and Vulcan were in the Klingon Empire, then Star Trek would parallel Poul Anderson's Flandry series. Many sf stories about spaceship crews exploring extrasolar planets could be adapted as Star Trek episodes.
Addendum: Are Moties like intelligent tribbles?