Should I spend time with my girlfriend?
I had to check to make sure Persona 4 followed the same pattern as 3 and 5, which I'm more familiar with, but it appears that it does. That is, you don't get anything for hanging out with your girlfriend (or any social link, for that matter) after they have hit max rank.
From here:
no, you don't get anything by spending time with them after maxing SLs.
However, it's worth mentioning that maxed social links will have an additional scene at the end of the game if you talk to them on the final day.