It would be nice if you got some kind of message in your chat window that
you just received a turn, especially when both players are online and doing their turns quickly.

Theoretically, it's possible.

But at the moment there is no data exchange 'tween game server and chat server and the latter just "doesn't know" about turns and other game stuff.

Another thing is easy to make - if a player has just made his turn and is trying to send it, a (private) message is being sent to his opponent.

But, I don't like this idea...

How about the client quietly connecting and checking every X minutes?