https://assetstore.unity.com/packages/tools/integration/android-google-sign-in-103961

This plugin exposes the Google Sign-In API within Unity. This is specifically intended to be used by Unity projects that require OAuth ID tokens or server auth codes.

Supports only Android build.

See Google Sign-In for Android for more information.

To authenticate you need create credentials on the API console for your application. The steps to do this are available on Google Sign-In for Android or as part of Firebase configuration. In order to access ID tokens or server auth codes, you also need to configure a web client ID.

This file contains the client-side information needed to use Google Sign-in. The details on how to do this are documented on the Developer website.

Once you have the configuration file, open it in a text editor. In the middle of the file you should see the oauth_client section:

      "oauth_client": [
        {
          "client_id": "411000067631-hmh4e210xxxxxxxxxx373t3icpju8ooi.apps.googleusercontent.com",
          "client_type": 3
        },
        {
          "client_id": "411000067631-udra361txxxxxxxxxx561o9u9hc0java.apps.googleusercontent.com",
          "client_type": 1,
          "android_info": {
            "package_name": "com.your.package.name.",
            "certificate_hash": "7ada045cccccccccc677a38c91474628d6c55d03"
          }
        }
      ]

There are 3 values you need for configuring your Unity project:

1. The Web client ID. This is needed for generating a server auth code for your backend server, or for generating an ID token. This is the client_id value for the oauth client with client_type == 3.
2. The package_name. The client entry with client_type == 1 is the Android client. The package_name must be entered in the Unity player settings.
3. The keystore used to sign your application. This is configured in the publishing settings of the Android Player properties in the Unity editor. This must be the same keystore used to generate the SHA1 fingerprint when creating the application on the console. NOTE: The configutation file does not reference the keystore, you need to keep track of this yourself.

Create a new Unity project and import the AndroidGoogleSignIn.unitypackage (latest version). This contains native code, C# Unity code needed to call the Google Sign-In API for Android.

1. call var googleSignInScript = AndroidGoogleSignIn.Init(this.gameObject); to initialize the script
2. call googleSignInScript.SignIn(WEB_CLIENT_ID, GoogleSuccessCallback, GoogleErrorCallback); passing your Web client ID, a success and an error callback methods

1. Under Build Settings, select Android as the target platform.
2. Set the package name in the player settings to the package_name you found in the configuration file.
3. Select the keystore file, the key alias, and passwords.
4. Resolve the Google Play Services SDK dependencies by selecting from the menu: Assets/Play Services Resolver/Android Resolver/Resolve. This will add the required .aar files to your project in Assets/Plugins/Android.

Follow the instructions to use Firebase Auth with Credentials on the Firebase developer website.

Make sure to copy the google-services.json and/or GoogleService-Info.plist to your Unity project.

Then to use Google SignIn with Firebase Auth, you need to request an ID token when authenticating. The steps are:

1. Configure Google SignIn to request an id token and set the web client id as described above.
2. Call SignIn() (or SignInSilently()).
3. When handling the response, use the ID token to create a Firebase Credential.
4. Call Firebase Auth method SignInWithCredential().

    ..
    var googleSignInScript = AndroidGoogleSignIn.Init(this.gameObject);
    googleSignInScript.SignIn(WEB_CLIENT_ID, GoogleSuccessCallback, GoogleErrorCallback);
    ..
    
    private void GoogleCallback(string token) {
        Firebase.Auth.Credential credential = Firebase.Auth.GoogleAuthProvider.GetCredential(token, null);
        FirebaseAuth.DefaultInstance.SignInWithCredentialAsync (credential).ContinueWith (authTask => {
          if (authTask.IsCanceled) {
            //..
          } else if (authTask.IsFaulted) {
            //..
          } else {
            FirebaseUser loggedInUser = (Task<FirebaseUser>)authTask).Result;
          }
        });
    }

Post questions to this Github project.