Firefox uses a single standard URL for searches, which doesn't include any parameter to select an account. When you are searching interactively through Google's site, how do you switch accounts for your searches? Maybe you can see a change in the URL that specifies which account to use.

(The idea behind this question is trying to construct a custom search engine URL you could use in place of the standard one.)

Thank for Your try, but no differences in URL((

Hmm, if the URL does not control which account Google uses, it might be controlled by a cookie. You may need to search directly from https://www.google.com/ to get it to work the way you want.