For a start, try doing a search on the opposite problem -- turning quiz notification emails and assignment notifications off. There are many posts from people about trying to stop this, so if you find one that similar to what you're looking to do, then working backwards to turn this feature on may help you.
Before I started using Moodle, apparently our 'Manager' role (which all my teachers were initially set up as) caused teachers to get notifications about every assignment submission, even from students not in their classes. I don't know how this was 'fixed' or turned off, but again, see if you can find such as post and then work backwards to get it turned on for your situation.