There is currently no way to completely opt out of notifications of people that reply to one of your topics, quote you in a post, or other similar actions.
I get that it's a nice feature to have, and am not asking for it to be removed, but an option to be able to disable the toast windows at the top of the browser; they get extremely annoying in my experience.

Voted yes but won’t happen
Already been stated that they have no plans to implement this, as the notifications on quotes is the early stages to allowing @user notifying.
I’m fine with notifications and all, but I think there should be some changes:
People who make large public topics such as https://forums.d2jsp.org/topic.php?t=34605212&f=115
Or https://forums.d2jsp.org/topic.php?t=77450883&f=115
Public discussion topics for example, may find themselves quite swarmed with notifications all too often.