Have a look. https://www.google.com/search?client=firefox-b-1-d&q=does+google+save+my+search+history Firefox is not a search provider. see screenshot

No, it did not solve my problem. This is a privacy issue.

For the search, I first login to Firefox. I understand that for a search, I am passed off to Google. I need to know if my credentials of my account are passed off to Google because of this search. Again, I first sign in to Firefox. Privacy issue. Don't want my info sent to Google. I hope that this makes it more clear.

Firefox doesn't have it's own search it uses the standard default install for it's search engine. So unless you have a set home page search engine the default is used. Unless you sign into Google and do searches it will have searches on your Google data.