Turn on do not disturb for single app
No - do not disturb is a system level function on macOS and iOS.

What you’d need to do is make an AppleScript or other tool to automate disabling notifications for that app and then possible hook that up to a timer app or trigger (or just remind yourself to toggle it with a keyboard shortcut when your other manual timer goes off).

So a third party app on macOS clearly could do this, it’s just not been written (or covered widely enough to be found) since there’s no API to make that app easy to write.