Okay, here we go.
No, there have been no gay characters or relationships on Star Trek. In the more than 40 years of Star Trek, not one single time have they even acknowledged that gay people exist among the human species. No regular character has ever mentioned having or wanting a same-sex relationship, even though by the 1-in-10 at least 4 of them should have.
The closest we have come are the following:
TNG "The Host" - a race called the Trill work with the crew of the Enterprise. Beverly Crusher falls in love with a male Trill, only to find that the Trill are really a small slug-like creature living inside the humanoid body, which is little more than a carry-case. When the male host body dies, the slug is transferred to a new female host, and Beverly breaks off the relationship, unable to handle it.
Why this is not gay - It's really more like transexualism - changing one's physical sex. If Beverly simply isn't interested in sex with women, that's fair enough, can't really argue.
TNG "The Outcast" - a race called the J'Naii, who are gender-neutral, work with the crew of the Enterprise. One of them confesses to secrets feelings of female-ness, which is against all the rules in their society. Any expression of sexual identity, of being "male" or "female" rather than the usual neuter, is ruthlessly stamped out. Once this individual is found out by their authorities, and despite strong support and even a rescue attempt by the crew of the Enterprise, this individual is sentenced to reprogramming and eventually foreswears any of her previous "female" feelings.
Why this is not gay - The J'Naii who feels female begins a relationship with William Riker - a male - thus actually showing a hetero relationship, not a homo one. The character is played by a woman, thus further diluting the message. Also, the ending smacks of the execrable "ex-gay" concept.
DS9 "Rejoined - Jadzia Dax, a Trill woman, encounters Lenara Khan. In previous lives, Dax and Khan's former hosts had been married. Now that they're in new hosts, they want to rekindle their relationship, but are forbidden from doing so by Trill custom. Both hosts are now female, but at no point is that even suggested as being a problem or an issue. Rather, the issue is reliving an old life. Punishment means exile, and no more host bodies. Dax is willing to take the risk, but Khan is not, so they go their separate ways, but not before a female-on-female kiss.
Why this is not gay - They're not really lesbians - they're former lovers who just happen to now both be in female bodies. And when they were lovers, they were a hetero couple. But this is certainly the closest we've come. Another character even wonders, "What's wrong with just letting people love whoever they want to love?"
VOY "Warlord" - female Ocampa Kes is telepathically taken over by male alien warlord Tieran. She kisses a male and flirts with a female.
Why this is not gay - Again, it's more like transgender.
VOY "Ashes to Ashes" - a female lower-decks crewmember is asked to dinner by Captain Kathryn Janeway. The crewmember wonders briefly whether this is a date.
Why this is not gay - Because it's not. It's just a business meeting over dinner. But at least she didn't run screaming down the corridors at the very thought.
ENT "Stigma" - Vulcan sub-commander T'Pol reveals she has a degenerative disease that results from mind-melding - a practise the current Vulcan regime considers abhorrent. Because they don't approve of the action or the people, they do nothing to try to cure the disease.
Why this is not gay - because it's really about the initial stages of AIDS hysteria in the 1980s - even though theshow wasn't broadcast until 2003, making it rather belated.