Get web app login token in flutter app using OAuth2

I have integrated the social login feature in my web app. When someone logs in using Google(OAuth2), it creates the user profile in my DB(basically register the user automatically) and provides the login token for my app to the end-user. I want to trigger the same flow using my mobile application built for Android and iOS using Flutter. Do I have to trigger the flow using the web browser, if that's the case then how can I get the login token back in my app??