This is in part conjecture based on the comments above as I worked with ICL not IBM the word ticket is linked to the history of job tickets, and in my days was often seen as cards in slots on the wall.
Back in the 50's IBM had a system for converting customer sales tickets (like those on clothing) into computer punch cards for input to a database for subsequent stock control and other queries. There was a dedicated device IBM 549: Ticket Converter. As described in A24-1010-0_IBM_Operators_Reference. However going further back in the archives, we find that in 1921 IBM's predecessor (Computing-Tabulating-Recording Company) acquired the Ticketograph Company of Chicago.
"The Ticketograph was a device for controlling job labor" ... "It also could be used to track the flow of important documents and files throughout an organization."
It in turn was based on earlier job tickets as developed from punching tickets "in and out" all based on the following root.
"Dr. Herman Hollerith conducts the first practical test of his tabulating system in recording and tabulating vital statistics for the Baltimore (Maryland) Department of Health. He would receive the first patents for his Electric Tabulating Machine in 1889. Hollerith will later form one of IBM's predecessor companies."
The question further askes about etymology of entomology (bugs) in computing and that mix introduces several other historic dates.
Mid 60's the IBM dedicated support systems were called ReTAIN and "delivered more than 5 decades of value to IBM and its clients" for support, however I have not found exactly what it replaced.
1954 Mass-production of computers, Thus without an earlier reference
(except bug and help desk staff, see below) I have to currently stop here and offer that as the potential point of
significant computer bug support tickets.
1945 Sept 9th Bug found in a relay and applied to mean some form of computer error or failure.
In May 1932 IBM president Thomas Watson, Sr., formally established an Education Department here are the 1935 graduating "Systems service women" [equivalent to help desk staff] assigned to IBM branch offices to assist salesmen in assessing customer requirements and to teach the customer's employees how to use their new IBM equipment.