A third party cookie is a cookie that has been set across different domains. I.e: you are on foo.com, and a javascript request to bar.com sets a cookie on the bar.com domain.
A first party cookie is set on the same domain. I.e: you are on foo.com, and the page responds with a cookie for foo.com
Google paid Firefox plenty of money.
If you're doing a Google search, wouldn't a Google cookie be first party...?
Do you have a specific example you could share?