I think it makes total sense that you would want to bond with your T over something that is important to you both. Asking her is just a way to get it out in the open so you can talk about it.

BTW, I can't imagine that your T wouldn't know that clients might look her up on Facebook. Mine apparently thought her Facebook profile was more locked down than it is, but she wasn't upset that I looked at it. But still, I think it's fine if you just ask about cats (or just pets in general) without mentioning that you've already seen pictures.