I don't think you need to be honest about the fact of your Googling. Stuff on social media is public and if people don't want information to be accessible to strangers, they need to take responsibility for not sharing personal information in world wide arenas. Therapists and their families are not a special case. If it's out there to be read, your reading of it is not an overstep.

In many ways, I think the Googling itself is a bit of a red herring. I would want to explore what was driving my need to Google. For me, my questions would include am I searching for connection and reassurance? Do I feel safer observing the family events of others because my own relations are overwhelming? What aspects of my internal world are trying to get my needs met by social media searching? And what is my guilt about? That I am not deserving of connection?

You can explore the underlying needs without exposing the extent of your searching. Therapeutically, the searching itself is probably the least important factor. You don't need to hide the level of your searching because it is wrong or something to be ashamed of (it's normal and expected, why else do people post stuff to be read?!). Nor do I think you should hide your actions out of fear (although I have no doubt that some therapists would not respond compassionately). You do not need to be 100% transparent, you are allowed to have private actions. Exploring our needs is the most important factor for clients.