Pretty unlikely. The ITA matrix site is basically there because by using it, we give them free testing for a product they sell to other people. It's not really publicized, and the "publicized" version is with google flights -- http://www.google.com/flights/ -- which is already aware of your sign in, although it doesn't require one.

I wouldn't be surprised if matrix actually does report back some information to Google if you're signed in, but you never know. I know people at Google on both sides of their travel search dept (ITA and the main google part) but they sure as heck won't tell me anything about things like that.