The following solution works in flutter project:

Step 1: Generate SHA1 and SHA256 keys.
Step 2: Add both the SHA1 and SHA256 to firebase. (in you app settings)
Step 3: Download google-services.json to android/app in your project folder.
Step 4: In your terminal run the command flutter clean .
Step 5: Run your flutter app.

For more information visit:
https://developers.google.com/android/guides/client-auth
https://firebase.google.com/docs/flutter/setup