IIRC, that was circulated BEFORE the movie was released, and may have been suggested by Paramount's publicity department.
IIRC, the rumour about a gay character being in a TNG movie happened first with "Generations". On UseNet and GEnie, fans were compiling all kinds of rumour mills about the movie - and someone claimed a new, young, male character would be featured and that, at some point in the movie, we'd realise he had a husband who was also a crewman.
Someone then claimed to have an early draft of the script and posted a snippet to an early BBS - but, in reality, it was a portion of David Gerrold's infamous "Blood and Fire" grafted into a real leaked section of "Generations".
When "First Contact" was announced, the rumour mill went crazy again, and - now that the early message boards permitted ease of photo sharing - early sketches of the Borg Queen surfaced, which turned out to be
real preliminary artwork. Someone else then recycled the above false script segment, this time using the name of Hawk - because a casting list had already mentioned the character name.
Finally, someone rustled up a photo of the actor, Neal McDonough, and this only inflamed the rumour more: "Hawk must be gay - he looks gay!", "He's too pretty to be in Starfleet", "He has gay eyes and gay hair", and other crazy lines.
Years later, gay activist/columnist/author and Trek fan, Andy Mangels,
reclaimed those old, false, gay rumours about "First Contact" and made the Hawk character gay in his ST novel, written with Michael A Martin, "Section 31: Rogue".