All the Tickets need to be permanently deleted from the system to be able to reset the count in your Helpdesk.
Delete all the existing tickets and then remove them from the Trash folder as well. Please click on the view name at the top of the page in the Tickets tab to access the Trash folder and remove the tickets permanently. The tickets in the Spam folders need to be deleted as well.
Make sure that there are no tickets so that the Ticket ID can be set to 1 from the next incoming ticket. Move to Admin > Helpdesk and set the next Ticket ID as '1'. However, make sure that this is done only if you have test tickets or dummy tickets in your account as the tickets cannot be retrieved once deleted. Also, you need to delete any tickets that are archived to be able to set the ticket count as '1'.