Had an issue for ages, and just found the fix, so thought I would share it in the event it will help others. This is based around the assumption you have got notifications set to "on" within the Connect App etc. and still not receiving notifications. Will also work for emails and any other notifications you are not receiving
On your watch, go to:
1. settings -
2. Phone -
3. Smart Notifications -
4. During Activity (choose your options on what comes through i.e. Show Calls and Texts or show all etc)
5. And this is where it got me, also go to "NOT DURING ACTIVITY" and chose "Show All".....Not during activity is the majority of your day.
Then you can go back to your Garmin Connect App - More (bottom left of your screen on Android) - Settings - Notifications - App notifications and choose what you want to come through on your watch. You will now get all notifications from the apps you chose to switch on from the connect App notification app menu
Hope this helped