Since Discord is an Electron app and functions like a webpage in every way, yup, you can also use it there.
Although testing it now, alt+control+down seems to move me to the next server, rather than channel. If that is what you are after, control+tab does the same thing, or you could use control+the number of the server you want to go to (with 1 being home). For navigating between channels, use alt+down or up, and for navigating between just the unread ones, alt+shift+down or up.
$1However, if you ever find you need NVDA to ignore a keypress elsewhere than a website/web application, try NVDA+f2. It will pass just the next key or combination you press through to the system.
If any of NVDA's shortcuts just bother you altogether, you can also change them! Press nvda+n and go to Preferences>Input Gestures. You can delete hotkeys there, change them, or even add them in some cases. For example, the article element in webpages does not have a dedicated hotkey, but I've set it to j and shift+j, which helps a lot on Mastodon and other such websites with plenty of them.