Try updating your browser to see if the issue would be solved

Can you connect to Google sites (search, Gmail) when you first start up Firefox and then it stops working after a while, or can your Firefox never connect to Google sites at all, ever?

If it stops working after a while, there is a setting you can change which might help with that. What you do is turn off the "zero round trip resumption" feature. Here's how:

(1) In a new tab, type or paste about:config in the address bar and press Enter/Return. Click the button accepting the risk.

More info on about:config: Configuration Editor for Firefox. The moderators would like us to remind you that changes made through this back door aren't fully supported and aren't guaranteed to continue working in the future.

(2) In the search box in the page, type or paste security.tls.enable_0rtt_data and pause while the list is filtered

(3) Double-click the preference to switch the value from true to false

After that, try Google again in a new tab. Any improvement?