I am using Google Chrome browser on my system. Before answering this question I just installed the latest version of Mozilla Firefox and checked it with several sites several times. But I couldn't find any such problem described here.
If you are having the above-said problem only with Google or certain sites I suggest you clear all browsing data including all cookies and everything. You can do it by going to the location -> top right button on Firefox -> history -> clear recent history -> then under Details tab check all checkboxes there. Most probably this may solve your problem.
If this procedure can't solve it then you must check your system for any malware with multiple tools. Do confirm you are using the latest version of firefox instead of a very old version. Also, check for any extensions or plugins in your browser like browser safe, browser protect, browser guard, site protect, secure safe or something like these names or anything suspicious. If there anything like that remove it. You may also check your system for such applications installed. This must solve your problem.
If you can't manually remove those applications or having any difficulty in removing such malware or adware you may use a software like Superantispyware. It is a free tool which helps you to scan adware, malware, viruses and malicious browser extensions.