Don’t “try” to get her to open up. Everyone has limits of what they want to tell people. Not every limit is the same; your best friend/partner/family don’t need to know everything about you in order to be “close”.

Respect her boundaries. Make sure she knows she can trust you not to push them. If she ever wants to open up about things she will. I say this as a closed off person myself; I HATE when people think they have to know every inch of my life to make them feel we are close. It’s really messed up and damaging to crack someone when they don’t want to be. “Friends” or people I’ve dated who have done that to me have now accidentally made me even more closed off than I was before. So, please, if she’s closed off don’t try and get her to open up. She will open up to you whenever she wants if she wants.