What you are describing [dependent lists] is not currently a feature in osTicket.
The only way to get around this that I know of (without altering the code) is to rename your Help Topics using your "Department" names.
ie goto Admin panel -> Manage -> Help Topics.
Make a Help Topic for each Department. IT, Finance, Maintenance, etc,
Then create lists for each Help Topic that would be a list of your Department's Help Topic categories.
ie go to: Admin panel -> Lists
Create a new list for IT. Add all entries to it for each "Help Topic" that you want.
Create a new list for Finance. Add all entries to it for each "Help Topic" that you want.
repeat for each actual Department.