Do you use any add-ons to enhance your privacy on Google? To make sure those also are the latest versions, try updating your extensions on the Add-ons page. Either:

• Ctrl+Shift+a
• orange Firefox button (or Tools menu) > Add-ons

In the left column, click Extensions. Then, above the list, click the "gear" button and "Check for Updates".

When you have a problem with one particular site, a good "first thing to try" is clearing your Firefox cache and deleting your saved cookies for the site.

(1) Bypass Firefox's Cache

Use Ctrl+Shift+r to reload the page fresh from the server.

Alternately, you also can clear Firefox's cache completely using:

orange Firefox button (or Tools menu) > Options > Advanced

On the Network mini-tab > Cached Web Content : "Clear Now"

If you have a large hard drive, this might take a few minutes.

(2) Remove the site's cookies (save any pending work first). While viewing a page on the site, try either:

• right-click and choose View Page Info > Security > "View Cookies"
• Alt+t (open the classic Tools menu) > Page Info > Security > "View Cookies"

In the dialog that opens, you can remove the site's cookies individually.

Then try reloading the page. Does that help?

Also, I believe Google requires third party cookies. This support article has the steps for making sure you are accepting website cookies: Websites say cookies are blocked - Unblock them.

This is probably off the point, but I use Google search logged out nearly all the time and never get bothered to log in. Is there any chance that some malware has snuck in and is trying to steal your login?

In case one of your extensions is involved, could you test the page in Firefox's Safe Mode? That's a standard diagnostic tool to deactivate extensions and some advanced features of Firefox. More info: Diagnose Firefox issues using Troubleshoot Mode.

You can restart Firefox in Safe Mode using

Help > Restart with Add-ons Disabled (Flash and other plugins still run)

In the dialog, click "Start in Safe Mode" (not Reset)

Do the prompts stop?

If you think it's relevant, this article lists scanners other Firefox users have found helpful: Troubleshoot Firefox issues caused by malware.

It shouldn't be necessary to enter a Google password to be able to search on Google.

Make sure to only enter such a password in a log in form on the Google log in page (accounts.google.com).