No you reset the auto increment to 1. So any new tickets will start at ticket_id 1. You seem to be confusing the ticket_id (internal reference) and ticket number (number, external reference)
If you want to change a ticket number of that ticket you would have to edit the number field. Then you would have to go to a bunch of other tables to change the reference to that ticket id to 1 also. Your best bet is to delete the ticket and re-create it after you do:
Go to Admin panel -> Settings -> tickets.
Click on Manage.
Change the next field to 1.