I don't think there's anything wrong with searching for info on our t's on the Internet, as I consider it all public info. But I agree with the others that wanting to be added to your t's Friends list on Facebook is not a good idea. I think it crosses boundaries. T's aren't permitted to have clients as friends, as that is considered having a "dual relationship," one in which you are therapist-client, but also friend-friend. It is consider unethical.

Consider talking to your t about your discovery of her Facebook page. Let her know you desire to know more about her life, and why. Perhaps she would agree to disclose a bit more to you about her life, within the boundaries of the therapeutic relationship, without crossing boundaries. Since the therapy relationship is so unique, and not like any other relationships we have in life, it might be helpful to ask your t to explain to you more clearly what the therapeutic relationship involves, what can be included in it, and where the lines get drawn.