Solution 1:

Discord also doesn't show up for me in System Preferences -> Notifications, however there are two places where Discord notification settings are, both accessible from the app.

From the top ribbon, click Discord > Preferences and there's a Notifications menu. From there you can disable Desktop notifications.

Additionally, if you right click each channel you're in you can see the per-channel notification settings.

Finally, in order to actually clear the badge after I turned notifications off, I right clicked each channel and clicked "Mark as read".

Solution 2:

It usually means you have an unread message in a channel. You can disable it by going to your System Preferences -> Notifications -> Discord -> Uncheck the "Badge app icon".

Notification Settings