I have heard the reason the movie brokeback mountain take place at Big Horn Mountains is because of the Goats that lives on the mountain actually are homosexual (sorry if I did not use the correct terms here). Is it just a rumor?
Very much a rumour.
From USA Today:
Coon [Wyoming Business Council spokesman] said Ang Lee, the director of Brokeback Mountain, toured much of the eastern Big Horn Mountains and several nearby towns when scouting locations for the film. But because of budget concerns, Lee shot the film in Canada.
Financial incentives have drawn many film companies to Canada, which has built a $5 billion film industry in the process. Because of the high amount of production there already, companies can hire local crews instead of bringing them from elsewhere, cheapening the overall price of the project.
It's also worth considering these two different responses by Ang Lee, the director, in interviews.
First, from MovieWeb:
The setting plays an important role in your films. This film is an epic cowboy movie set in Wyoming. Was this another thing that sold you aside from the romance?
Ang Lee: Yes, I think that sold me and helped the romance. I think great romance needs great obstacles and textures. Romance and love are abstract ideas, an illusion. How do you make that? I think, most of the time, obstacles help build the romance. It helps to envision and make it feel real to you. I think that mixture is ultimately very interesting because they're very macho, but romance is usually soft. That strange mixture was very fresh and helped me to grope into what love is.
Second, from AboutFilm:
Question: You traveled with [screenwriter] Larry McMurtry to Western locations while scouting the movie. What surprised you about Western people or locations that you didn't know before?
Lee: The location, we have seen it. We have traveled through it or [seen it in] a movie. I think it was more the people. Of course, they are always nice people. They are just like everybody here, except they are nicer, and I felt guilty that I was going to do a tough movie about them. [Laughs.] Something struck me, [with my] fresh eyes. Sorry to say—[the] eccentric[ity]. You go to a bar and just see the things that they hang up. I can't believe the things they hang up inside and outside of bars. So imaginary. I don't know how else to put it. Of course that will annoy a lot of Westerners, because [they'll say], “We have a lot of normal people.” There are two sides. One is like that. It's like [they] don't want [you] to go into their territory. I saw a sign in Wyoming—a “No Trespassing” sign. And it says, “Trespassers will be shot. Survivors will be shot again.” [Laughs.] Some characters are like that. If they hear you're coming with the Wyoming film commissioners, [they want] nothing to do with the state.
But they are also very nice people, because they are lonely and when you go to the countryside and want [friendliness], you can't shut them down. They keep talking and talking and talking, and they have this Wyoming lazy town kind of thing. You have to pay a lot of attention. They have this easy tone. Very slow, very easy. Relax, and we'll talk about something really tragic. One tragic story after another—the most miserable things. It's just on and on, slowly. And you have to pay attention. They take long pauses. So, I learned a lot like that. Of course, [if] you watch Larry McMurtry you learn. He's the epitome.
These suggest the primary reason for filming there is that it had beautiful locations, friendly people and was a traditional "macho", cowboy area - a perfect backdrop considering the story.