Depends what you mean.

Firefox uses a secure connection: searches are sent using HTTPS, which prevents anyone listening on your connection from reading what you and Google are exchanging.

On Firefox's Side: Privacy questions

Firefox stores URL history: anyone who gains access to your computer can see your searches in history. If you want to search without Firefox storing results page URLs, you can use a private window (Ctrl+Shift+P).

Firefox stores your search terms in search history: terms from search history show up with a little clock icon in the address bar drop-down while conducting future searches. If you want to prevent Firefox from storing or displaying search history, please let us know and someone can provide the steps for that.

On Google's Side: Privacy questions

Google can store your search request and actions you take on the results page (similar to any website you interact with). You can search logged out of Google to prevent requests from being accumulated in your Google account. There may be other ways to be more "anonymous" when using Google, but I haven't researched it.

Maybe I asked the wrong question? Try this: If I do a search using google (search bar) and google returns results (in Firefox), than I click on the links provided (hits) -- does google track my browsing from "that point on"?

Of course they track you.

When you click a link, Google often records that action. There is an add-on to remove the tracking script from results page links: https://addons.mozilla.org/firefox/addon/dont-track-me-google1/

After that, unless the site you're visiting embeds Google content, Google does not have visibility on where you go from there.

I use Firefox exclusively! Considering why people use Firefox (security and I just like it) - Maybe mozilla would consider developing their own Search Engine?

I use Firefox exclusively! Considering why people use Firefox (security and I just like it) - Maybe mozilla would consider developing their own Search Engine?

It's very expensive. You could experiment with DuckDuckGo (one of the built-in options), which promises more privacy.

As I like to say: You got something better to do?