No. Google would not take the right to do that (I don't think any browser has done such). Also many "hidden" pages would anyway not be accessible; i.e. I may have many pages on my computers that are 100% private; also many websites have pages that are not accessible unless you first create an account and log in.
However, would Google be able to eventually find the page? Yes.
- If you have links on that page to lead to other public pages and you allow for the Referer¹, then your page URL is now out in the wild. Are referrer URL made public? Pretty rarely, but I've seen such happen in the past, so it can happen.
- If you tell enough people where the page is, eventually someone will make that URL public. i.e. post the URL on a forum, send it to someone else via email and that email becomes public, etc.
- There are also bots that will try, at least seemingly, random paths against websites. They could inadvertently discover your page that way. Those bots may then create a page on their owner's website with a link to your hidden page. I've seen such strange websites which attempt to grade your page in what at least to me looks like totally random algorithms.
You also mentioned Google Analytics. It would be interested to test (verify) that they do not share the URLs they get in Google Analytics with Google Search. They also are not supposed to do that either.
However, there is the Alexa toolbar. This one's from Amazon.com (I'm sure there are others similar toolbars). It adds a script to your browser which essentially sends all the URLs that you access to Alexa. Probably a bad idea. So such exists, but it's not in the browsers by default.
¹ the spelling of "Referer" is wrong in the HTTP headers and it was never fixed. Computers don't care much.