Stop full screen alerts on my Apple Watch
The closest feature to what you're describing is called Notification Privacy, and it lets you get a quick summary when you twist your wrist instead of the full notification. From the Apple Watch user guide:


  When you raise your wrist to see a notification, you get a quick summary,
  then full details a few seconds later. For example, when a message arrives, you see who it’s from first, then the message appears. To stop the full notification from appearing unless you tap it, open the Apple Watch app on iPhone, tap My Watch, tap Notifications, then turn on Notification Privacy.


Otherwise, you can turn off Apple Watch notifications for specific apps in the Apple Watch app.