Do professors ever befriend former students?
First, if you are ever again going to be in one of this professor's classes, then such a friendship is unlikely, at least until you are out of school.
If there's no chance that you will ever again be in a student / professor relationship, talk to her.  Say what you wrote: &quot;It would be lovely to have a professor I could talk to and look up to as I move through grad school.&quot;