To explain a little why we want this:

We get automatic alert messages to our support email about the state of our product's servers (when they are down or experiencing a fault, for example). We want them to be in Desk so the information is readily accessible, so we created a department for them, but we don't really want to be getting the email notification every time they arrive.

Is there a way to turn that off? We only want it turned off for one department.

The notification settings that we can toggle don't include this notification as it still fires when all of them are off.