You don't have to find him even if your therapist doesn't tell you.

I wonder, where does this need to know come from? Are you aware what the reason for it is? What does it mean for the process you and your therapist are involved in together? What might be its significance in your life? That would probably be very useful for you to be aware of, whether or not you look this guy up. (You don't need to answer me, I'm just saying this because I personally find questions useful to help me think and decide in difficult situations).

We're all different and it's up to you what to do. I just want to say though, without knowing all the details (so I may be wrong), but maybe your therapist has a very good reason not to tell you who her boyfriend is, even beside her own personal boundaries. I got to know my therapist's ex girlfriend when they were together and it caused me a whole lot of issues in my therapy, in my studies, in my private life. I learned a lot from my difficulties but I think my therapy would have been a safer space for me had I not known who his girlfriend was.

Now your experience will probably be different. I'm just saying, sometimes, even if not knowing may trigger a lot of curiosity and fantasies, knowing can be worse in a very intense long-term therapy relationship, like mine was and like yours also seems to be. So please keep that in mind too when you decide whether or not to look him up.