I am a diehard Trekkie who's seen almost every last hour of Star Trek ever produced, but even I wouldn't tell you to watch hundreds upon hundreds of hours of the show since it's not really something YOU like. I say take the advice of those who've mentioned Trek's best episodes and movies and see them. Or see the series she likes. But my advice is to get an appreciation for Trek, and a frame of reference for your gf's interests, but don't put yourself through it all.
Movies: Star Trek '09, 2-4, 6, First Contact
TNG seasons 3-6
DS9 seasons 3-6
In that order. If you like them, fill them out with the rest of it.
And I suggest these without including some of my personal favorite seasons and movies.