This is not going to work.
You know a webserver is always supposed to be online. That way, you can always make a request.
A client-side (user), does not need, and isn't always online. If they close the browser window, you can't interact with them anymore. The server would have a hard time checking what user just went down, out of the thousands of users you have.
Making a call to a webserver is possible, but making a call to a client is not.
There are, next to that reason, security issues if you would be able to make a request to the client-side.
What you should probably do, is make a seperate request for the authentication token from Javascript, make Laravel call the Firebase AUTH server and then return the token it received.
update
It is possible to make requests to a client using HTTP2, by using server-push. To do that properly, especially with an authentication token, you will need some experience in building webservices and security.