Yes
but, it depends on what you use for OBS as yoru audio source
If you use Desktop audio, and that includes Discord, then obviously you'd stream Discord audio
So the question, per your title, is how to do that. I don't game and I don't use Discord, so hopefully you'll get better replies. In the interim, I disabled default Desktop audio entirely, then make sure I specify exactly which audio sources I want in OBS... then you simply wouldn't add Discord into that mix... but ymmv depending on game audio capture details