I am using google authorization through firebase and it works properly also but what should I do to get the credentials of the user to the firebase real-time database as soon as the user logs in to the website.
There is no way for your web site to get the user's credentials from Google Authentication. That would be a huge security risk.
There is also typically no need to store the user's credentials in your own database. Firebase already stores the user's credentials (securely) on the client itself, and restores those when the page reloads.