It is something that causes one to have pause. Did she tell you prior to you Googling her that she didn’t do it? In other words, did she lie about it or was she straight up about it? It’s a big difference.
I have 2 opposite ends of the spectrum I have dealt with in my life. I had my step-daughter’s old boyfriend (who we didn’t like). He told me he had a criminal background, that he had gotten arrested for a minor drug charge – possession of marijuana. When I dug deeper, marijuana wasn’t involved at all. He had been arrested for manufacturing crystal meth. He only avoided heavy prison time by agreeing to testify against the others involved. He obviously lied, that made his status as a potential quality human go way down. On the other end of the spectrum, I had an applicant for a driver’s job that I interviewed. As a matter of the hiring process we have to do a criminal background check. I always ask the applicant if there is anything in their background that is going to show up. I tell them that if they say there is and we document it, that probably helps them more than not telling me and us finding out. He said there was. He had gotten out of prison just a couple years before. He was in prison for conspiracy to commit murder. It was a situation when he was a kid, running with the wrong crowd. They killed someone and because he didn’t turn them in, when they were caught, he went down with them. We ran the background check and that was it. He presented himself as a reformed individual with a wife and child and he was just looking to put his past behind him. I hired him and he ended up being one of the best workers, most upright and conscientious individuals I have ever met.
The point is that people can do horrible things and repent of them. They can try turning their lives around and become great people. Personally whether you talk to her about it or not is really up to you. If she has given you no reason to distrust her, you could go on with life without asking her at all. I think the hardest part is that you Googled her. That makes the conversation a bit more sticky. “Hey, I Googled you and saw you were a bank robber! How’d that work out for you?” Googling looks like you were prying. I think you’d have to be cautious about how you broached it. If you have heard she had a colorful past you could ask about it…say you heard she had some rough patches in her past and ask what went on.
If she is now a good person, I would let it lie. If you grow closer to her, it is likely she will tell you about it. If not, does it really matter?
Just be cautious if she ever asks you for a ride to the bank!