FaceTime is great and works really well for photo and video calls across a number of devices. However, it can be annoying to get alerts for on your Mac.
Turn Off FaceTime on Mac
It is not immediately obvious how you can turn off FaceTime on your Mac. There are actually a couple of ways. The first is to click FaceTime > Turn FaceTime Off.
Next time you want to use FaceTime on your Mac, open the app, and click Turn On. You can also Sign Out completely. Go to Facetime > Preferences > Sign Out.
You will have to log-in using your AppleID next time you want to use the app. Doing either of the previous steps does not stop you from getting FaceTime alerts on your other devices such as an Phone or iPad.