On a first date I brought a girl to my apartment and booted up Donkey Konga on the GameCube. She played it. She said she liked it. A few years later she married me.
DO NOT TAKE THIS AS AN ENDORSEMENT
Because now whenever anyone asks her about her first date she tells them the story of how I brought her to my apartment and brought out Bongo Controllers. Her friends all laugh at me. I feel ashamed just writing about it now. I was young and since this girl had mentioned that she likes game a bit I thought it was a good idea.
It is not a good idea.
She only said she liked games to try to impress me she only put up with them because she liked other qualities I had. Once we got farther in the relationship she was comfortable enough to tell me I almost blew it that first night.
I am 99% sure the girl you are talking about is not a gamer hoping to see you collection. If she is, go for it, but let her make the first move. Don't bring the games out, don't suggest one unless she reminds you that her reason for being with you, alone in your apartment is to play video games.
As a happy ending, my wife and I do now play games together from time to time. We beat Super Mario 3D word together and she likes Captain Toad. She can also probably beat most of us at Tetris. But that's about it.
My 2 cents.
EDIT: And I didn't look at the date.