So ... you have used two different osTicket terms. These terms are not really compatible.
You talk about having Teams, and then ask about Departments. So which is it?
You can turn on ticket alerts and notices at: Admin panel -> Settings -> Tickets -> Alerts and Notices.
If you do not want a specific Department to get Alerts and Notices, then you can go to Admin panel -> Agents -> Departments and disable them. Click on your Department and see Alerts and Notices
Teams (which are cross departmental groups) do not have this functionality.