I learnt to ski as an adult in my easrly 30s.
I went on holiday to Europe with my partner and his family (I'm in the UK). During the week we were there, I had 4 half-day lessons, and then I would spend the rest of the time skiing with my partner and his family.
My partner at the time (we're no longer together) was kind of impatient and his expectations were somewhat unrealistic. He would disappointed that I still wasn't fully parallel at the end of that week (I could make parallel turns, but tended to revert to stemming turns when I was nervous), that I was still getting nervous on steeper sections of blues and that I didn't really want to get on reds (roughly hard blue or easy black diamon), and he was also made it clear that he thought I was too slow. If I had been there to try "his" sport, I probably would have given up, but I also wanted to learn to ski for myself too, so I stuck at it.
His family, on the other hand, were great. They would take time to ski with me on easier runs, to let me go at my own pace, and to encourage me. His brother would sometimes just slow down and let me follow his path when I was nervous, which was a much better approach for me than trying to give me tips.
Most of the advice given here fits well with my experience - be patient, let your girlfriend choose her own pace and don't try to teach her. When you're out with her, focus on fun rather than objectives, and only give tips if they are asked for.
I also wanted to say that saying "don't take her into difficult terrain" sounds obvious, but it does sometimes happen through misjudgement (as you said, all greens look flat when you get more advanced, or she may misjudge something through inexperience). In those situations, it was always important to me to retain my autonomy. I'm ok with people giving suggestions, e.g., to suggest that a particular line might be easier, but ultimately making decisions about how to proceed for myself helped me to feel more in control.