I'm not going to say that this is impossible, but integrating Google Authenticator would be a nightmare, and likely wouldn't do what you're expecting it to do. Allow me to explain.
Problem #1
For each instance of the app, you'll have to have a Unique Identifier to pair it to Google's Authenticator. In other words, you'll need to generate a new QR code for each device, pass that off to the system (Which doesn't exist without the app.) to pair the application. This is going to require a "Log-In" to properly store.
Check out this for a possible work-around to Problem #1: https://authenticatorapi.com/
Problem #2
If you're already logging in and the device is already being authenticated, what purpose does the Google Authenticator provide? Well, I'm assuming it allows you to prevent unauthorized access, possibly prevent more than one device. There are infinitely easier ways to manage this, location services are the first one that comes to mind for me.
Possible Solution (iOS)
This is absolutely my go-to method for handling authentication. Encourage or require your users to use 2FA (Two Factor Authentication) attached to SMS messaging. A simple line of code such as this will grab an SMS one-time code.
// Available as of iOS 12, make sure to check version.
self.verifyCodeTextField.textContentType = .oneTimeCode
Android SMS Retrieval
Android Auto-SMS-Retrieval https://developers.google.com/identity/sms-retriever/overview
I don't know it all, most certainly, but I hope this opens the floor for some discussion and brings forth a solution or explanation to your question.
Research Material
Google Authenticator available as a public service?
java API for google Authenticator