How do I get started with Google Firebase?

Do you mean in the app, or elsewhere a button is pressed?

You can setup your app to receive notifications from Firebase in the background

You would use the native firebase component in AI2 to do this, or you can use the web component.

https://ai2.metricrat.co.uk/guides/firebase-with-a-web-component

You can search this community for information on how to work with an ESP8266.