The App Store won't ask for your Google Password at all for any reason. Apple and Google are separate companies and one does not have access to the accounts of the other to be requesting passwords for it.
The Starz app may ask for your Google password only if you are using your Google account for the subscription to it.
What exactly is asking for the password? Are you sure it wants your Google account password it's asking for?
Are you using your gmail email as your Apple Account? In that case, it may be asking for your Apple Account password not Google, and while they can be the same password if you set it up that way, they are not tied or linked together in any way.