Integrating Google Sign-In

Firstly add google_sign_in dependencies in your pubspac.yaml file

Then add internet permission in your androidManifest.xml file

Then write your code in main.dart

import'package:flutter/material.dart';
import 'package:google_sign_in/google_sign_in.dart';


GoogleSignIn _googleSignIn = GoogleSignIn(scopes: ['profile', 'email']);

void main() => runApp(MaterialApp(
  debugShowCheckedModeBanner: false,
  title: 'Google Sign in',
  home: SignInDemo(),
));

class SignInDemo extends StatefulWidget {
  @override
  _SignInDemoState createState() => _SignInDemoState();
}

class _SignInDemoState extends State<SignInDemo> {
  GoogleSignInAccount _currentUser;

  @override
  void initState() {
    super.initState();
    _googleSignIn.onCurrentUserChanged.listen((GoogleSignInAccount account){
      setState(() {
        _currentUser = account;
      });
    });
    _googleSignIn.signInSilently();
  }
  @override
  Widget build(BuildContext context) {
    return Scaffold(
      body: Center(child: _buildBody()),
    );
  }

  Widget _buildBody() {
    if (_currentUser != null) {
      return Column(
        mainAxisAlignment: MainAxisAlignment.center,
        crossAxisAlignment: CrossAxisAlignment.center,
        mainAxisSize: MainAxisSize.max,
        children: <Widget>[
          ListTile(
            leading: GoogleUserCircleAvatar(
              identity: _currentUser,
            ),
            title: Text(_currentUser.displayName ?? ''),
            subtitle: Text(_currentUser.email ?? ''),
          ),
          RaisedButton(
            onPressed: _handleSignOut,
            child: Text('SIGN OUT'),
          )
        ],
      );
    }
    else{
      return Column(
        mainAxisAlignment: MainAxisAlignment.center,
        crossAxisAlignment: CrossAxisAlignment.center,
        mainAxisSize: MainAxisSize.max,
        children: <Widget>[
          RaisedButton(
          onPressed: _handleSignIn,
          child: Image.network('https://www.pngarts.com/files/4/Google-PNG-Image-Transparent-Background.png',
          height: 70.0,width: 70.0,),
            shape: CircleBorder(),
          ),
        ],
      );
    }
  }

  Future<void> _handleSignIn() async{
    try{
      await _googleSignIn.signIn();
    }catch(error){
      print(error);
    }
  }

  Future<void> _handleSignOut() async{
    _googleSignIn.disconnect();
  }
}

Confir your project in firebase and google peaple api