Some support teams never delete tickets; they instead move them through their workflow to close them using an automation. You can however delete a ticket if you want. When you do, it is immediately deleted and cannot be recovered.
Zendesk Classic: The following procedure is not applicable to Zendesk Classic. To delete a ticket in Classic, select the ticket update option Delete ticket. Click OK in the warning, then click Submit.
To delete a ticket
Open the ticket that you want to delete.
Click Ticket options in the bottom toolbar, then select Delete.
You'll be warned that the ticket will be permanently deleted.