Is there a way to load async data on InitState method?

You can create an async method and call it inside your initState

@override
void initState () {
  super.initState();
  WidgetsBinding.instance.addPostFrameCallback((_){
    _asyncMethod();
  });
        
}

_asyncMethod() async {
  _googleSignIn.onCurrentUserChanged.listen((GoogleSignInAccount account)     {
    setState(() {
      _currentUser = account;
    });
  });
  _googleSignIn.signInSilently();
}