I've just set this up for an account for someone, and I'll go through the whole process (I expect you've already done some of this)
Once you're signed in to a Google service, go to the "My Account" option, and in the "Sign in and security" section, choose "Signing in to google"
You'll then see the option to enable 2 step authentication, so set this up if you haven't already (this will require confirming the phone number is accurate by receiving an SMS).
Once you've done this (or if you go back in to the 2 step authentication section), you'll see the advert to add Google Prompt at the top, below that is your current 2nd step authentication method, and then in the section below, are the options for "Set up alternative second step":
You want the third option to use the Authenticator app, and when it asks if you've got an Android or iPhone, choose Android - this gives you the QR code that the Microsoft Authenticator app will understand and work with.
Once confirmed, Google will automatically change to using that method