This Seeking Tech guide will explain how you can turn off trophy notifications and screenshots on the PS4.
1. To disable trophy notifications, go to the Settings menu.
2. From there, head to Notifications.
3. You should then select Pop-Up Notifications.
4. Afterwards, make sure that the Trophies option is turned off.
5. To turn off trophy screenshots, select Sharing and Broadcasts from the Settings menu.
6. Next, you should select Screenshot Settings.
7. You should then make sure you disable Take Screenshot Automatically When You Earn A Trophy.