Can I change a 15a breaker with a 20a breaker?
No, you can't just "upsize a breaker".  The breaker has a job, and you would break its ability to do that job.  I gather you don't understand what that is, but fooling around with creaky old wiring... Well, I'm glad you asked first.  

No, take the breaker back to the store.  The only breaker you should be installing is a 15 amp AFCI. Where you have old/suspect wiring, AFCI is a fantastic safety system. 

Anyway, that won't fix your problem

Breaker size is not the issue.  Bathroom fans are considerably less than 1 amp, they certainly shouldn't be tripping a 15A breaker, for Pete's sake.  Your fan has a big problem.  It's not supposed to do that at all.  

So either the fan has been miswired, or, the fan is defective. 

Maybe it's time to call a competent handyman or electrician.