yes, firefox has various business ties with google. amongst others they are installed as a default search engine in firefox in many parts of the world and provide filter lists for blocking malware and phishing domains ("safebrowsing service").

I would be remiss if I did not mention that your Firefox identified itself to the forum as version 43. An update is available: this article describes how to get Firefox 54 using the "About Firefox" dialog on the help menu: [[Update Firefox to the latest version]]. If you need to use legacy plugins other than Flash, see this article about using the Extended Support Release of Firefox 52 instead: [[Why do Java, Silverlight, Adobe Acrobat and other plugins no longer work?]] If something is holding you back from upgrading, please let us know so we can suggest solutions or workarounds. Version 43 is not secure; Mozilla discloses [https://www.mozilla.org/security/known-vulnerabilities/firefox.html security flaws] after each new release. Sometimes Firefox reports the wrong version because that information was saved in a preferences file. In that case, see: [[How to reset the default user agent on Firefox]].