Can I turn off email notifications for new chat messages?

There is no general setting to block all notifications for chats.

If you want to disable notifications for a chat, first click on the chat icon.

Now click on the chat for which you want to disable notifications.

Now click on the three dots in the top right corner.

Now click on Mute notifications.

The chat is now muted. You can recognize a muted chat by the crossed-out bell icon.