Burt,
When I first met my wife about 20 years ago, she said "no guns, no motorcycles, and don't even think of being a cop".
Today, she owns her own gun, rides on the back of the bike on long trips, and I'm a member of our volunteer Sheriffs Reserve Uniformed Division.
How you ask? Patience, love and understanding. As foreign as it is for us that someone would not want to protect themselves, it's as foreign to them that we would want to "tote around a gun".
I left "Armed Citizen" pages open from the NRA magazines, I'd leave my EDC pistol on the end table between us when watching TV. She knew it was a part of my life and generally when anyone sees a firearm, they are naturally curious. She used to stare at the pistol every once in awhile and even at times asked a particular question about it. I would mute the TV, unload the weapon, show her it was unloaded and then answer her question (I would not go into any more detail unless I could see she was still curious). By following safe gun handling rules and being patient with her, eventualllllllllllllly I had the conversation that would be received well by her. The conversation went something like this... "Hon, it would just about crush me if something ever happened to you in the house when I was not there. Maybe you should consider at least learning how to shoot." From there it still took some time, but she eventually went to a "Girls and Guns" class sponsored by the NRA. I don't know about your relationship, but I don't think it would have worked out well if I was her first instructor. She is an excellent shot now, has her own pistol, and a license to carry.
I feel much better that she can protect herself when she has to. I know she wouldn't be where she is now with firearms if I pushed it.
Only you know your girlfriend. Hopefully, my experience may help you.