If you use Google from a public computer like in a public library, and you don't log in to a Google account such as Gmail, nothing will be logged associated with your personal search history, only with the history of the public library computer.
If you're logged in to Gmail etc., the information you search for will be registered to your account search history, and the IP of the library computer will be associated with your account e.g. "Places I've been." Google will know you use that library. It may care that you searched for those things at the library and not at home, but I doubt it. But it's possible.
This is why you really don't want to check your e-mail from the public library. Or from any computer you don't control for the majority of the time. Even if you're using an incognito window, or the library has protections built in to segretate session data between individual library patrons, your info is on that computer. Another user could conceivably see what you searched for. You can't guarantee it won't happen, so in the security world, that's the same as "it could happen, and you should assume it will."
Just use Google without logging in to Gmail/Google docs/etc on that machine. Or even better, use duckduckgo.com ....
Hope this helps.