So it sounds like this girl has the same interest in games as you do. Is that right? If so, at least you know you have something in common. I would start by asking her about that. I would say something like "I've seen you in here a few times before. How long have you been playing ___________". Then go on from there. Ask her how she got in to playing that game and what other games she's interested in. Just start it off with something like that. If she seems to be friendly and responding to you positively (smiling, asking you questions back), just move on from there. Like maybe, "Aside from hanging out around here, what else do you like to do with your free time". And just keep going on from there to talk about music or whatever else. Just be friendly and try not to sound like you're badgering her for information. When you do talk to her, let us know how it goes. I'm curious now Good luck, but if it doesn't go that great, that's fine, too. I think you'll be fine, though.
By the way, my boyfriend is 23 and a big a computer geek. He has recently started getting back in to tabletops like Warhammer 40k and old school board games like Risk and Diplomacy, but he has never played Dungeons and Dragons. I'm not so much into "geek culture" (although I do have some Warhammer Orc figures that i've assembled and need to paint), but I tell him he's not a true geek until he's played D & D