osTicket does not have the concept of "approved tickets".
You could create some custom fields on the ticket, and then have a wierd work flow.
New tickets are created in Department - Approval 1.
Approver Agent 1 edits ticket inline to add something to the custom field approval1.
Approver Agent 1 transfers ticket to Department - Approval 2.
Approver Agent 2 edits the ticket inline to add something to the custom field approval2.
Approver Agent 2 transfers ticket to another department for processing by technicians.|