The simple answer to this question is no, at least within the movies and TV world. There are certainly openly gay characters within the associated Star Trek books.
There is a vast amount of material that can be read on this topic however. See Sexuality in Star Trek for a good overview of the topic.
Some of the producers (perhaps even Roddenberry) wanted to depict same-sex relationships to further enforce the "openness" of Humanity in the 23rd/24th centuries. The networks did not approve, even when the on-screen interactions were to be limited to background shots. Within the novels, these relationships are more often described. In the SCE series, one of the male crewmembers frequently writes letters to his beloved; another human male serving aboard a Starbase somewhere.
Hasn't it kind of been established that Sulu announced himself as gay in ST:TOS "The Way to Eden"?
^^ No. Sulu's sexuality was never discussed.
In the novelization of First Contact, it's stated that Lieutenant Hawk was homosexual.