Users are not in Departments. They can be put in Organizations.
Users also cannot open a ticket in a specific Department (unless you make Help Topics that route the ticket to the specific Department).
Agents can be in Departments (well they have access to them). Agents can open a ticket in any Department they want. You would make sure that the Department did not suppress New Ticket Alerts. And make sure that the Admin panel -> Tickets -> Alerts and Notices -> New Ticket Alert is enabled, and set to "Department Members". Then when a new Ticket is created it alerts all members.