There is nothing wrong with bringing up someone’s stutter and I would hope your girlfriend would be open to talking about it. I would question the tips and improvements that you intend to suggest and how much you actually understand about the condition. It is helpful when someone asks me if there is something they can do to help. It is a bit exasperating when someone chucks in some unwanted tips and tricks when really they have no idea what they are talking about. Generally speaking, the longer someone stutters, the more likely it is that they will continue to stutter. My girlfriend gives me confidence, backs me up when I get properly stuck and really isn’t bothered at all about my speech. I think this is the happy place to be. Make each other more comfortable and confident and see what happens after that.