Firefox is just a browser so has no control over what the search engine searches for or finds. Firefox is just "displaying" the google page, not controlling what Google finds.
All search engines have options to set the search level to various settings like "Safe" "Moderate" or "No filter".
Do a search with google (any search)
When the search results are displayed you see on the right a small "box" with a gear wheel inside it.
Select on this gear wheel and select "Search settings".
You see an option "SafeSearch Filters" which can be set as "No filtering", Moderate or Strict.
Set is as Strict and you should find you get less dodgy sites.



