Hi tuttifruttidev, is the situation as follows:

You start typing a query in Firefox's address bar, the top line of the drop-down shows what you typed, and "Search with Google" but then the results come up on a different site? Or does it already show a different site is going to be used for the search?

Either way, if it's not your choice, you shouldn't tolerate it. Here are two things to check:

(1) Clean up questionable extensions

One possible culprit would be an extension. You can view, disable, and often remove unwanted or unknown extensions on the Add-ons page. Either:

• Ctrl+Shift+a (Mac: Command+Shift+a)
• "3-bar" menu button (or Tools menu) > Add-ons
• type or paste about:addons in the address bar and press Enter/Return

In the left column of the Add-ons page, click Extensions. On the right side, find the "Manage Your Extensions" heading. If you don't have that heading, skip to #2.

Then cast a critical eye over the list below that heading. Any extensions Firefox installs for built-in features are hidden from this page, so everything listed here is your choice (and your responsibility) to manage. Anything that mentions search, or privacy, or incognito, or looks suspicious, or anything you just do not remember installing or why? If in doubt, disable (or remove). For your privacy and security, do NOT let mystery programs linger here.

Any improvement?

(2) Clean up your search engine settings

Firefox's Options page has a Search panel where you can set your default search engine -- the one Firefox will use when you start typing a query in the address bar -- as well as remove unwanted search engines. This article has the full details: Change your default search settings in Firefox.

If you see anything about "your organization" managing your Firefox, and no IT department is around, that can be caused by external administrative files which require a different kind of cleanup.

Any progress so far?

Finally, if things are showing up that you can't recall ever installing, it may be due to external malware. We have an article on clean-up tools here: Troubleshoot Firefox issues caused by malware.

You may have ad/mal-ware. Further information can be found in this article;
https://support.mozilla.org/en-US/kb/troubleshoot-firefox-issues-caused-malware?cache=no

Run most or all of the listed malware scanners. Each works differently.
If one program misses something, another may pick it up.