No. Google Messages only talks to Google’s servers.

The way it’s supposed to work is Google has some servers and an app, and you have your own servers and own app, and when someone sends a message from your app it goes your app > your server > Google server > Google app. And the same in reverse.

But in reality Google doesn’t let anybody’s servers talk to theirs and tries to make everybody use their app and their servers. Which they can afford to do because they own Android and can shove their app down anybody’s throat.