You can quit google without it affecting you in any way. And you should quit Google.

But since you probably have an Android phone (without an official Firefox OS port) you are surrounded on all sides by Google. But since Google took help from everyone in making Android a success by making it "open source", we have CyanogenMod, Android Open Source Project (AOSP), etc coming to our rescue.

Here is how you get rid of Google from your phone.

Easy way - which does not work

Slightly hard way - easily reversible - incomplete

Harder way - retains android - more complete

Even hard

Find a FirefoxOS build for your device or port one. Flash that.