I have hacked together a watcher for the Google Music website as a tampermonkey/greasemonkey script. It’s largely based on the chrome extension and uses the same “currently-playing” event that the Spotify watcher uses.
The script itself can be downloaded here:
https://git.zyrenth.com/kabili/user-scripts/raw/branch/master/ActivityWatch%20-%20Google%20Music.user.js