Taking notes from some other apps, Discord appears to be adding support for changing the app’s icon as a part of its Nitro subscription. As spotted by the Discord Previews account on Twitter/X ...

Note: Unfortunately, there's no dedicated Discord app for PlayStation consoles. This means you can't chat from your PlayStation; ... To do this on a computer, click the gear icon in the lower-left.

Open the Discord app. If you're on mobile, tap the hamburger icon in the top right corner ("≡"), then tap the profile icon in the bottom right corner of the screen.