Try the attached report. You'll need to rename it to .report so you can import it.
The only way I know how to associate tickets to groups is via the user. In other words a user is a contact for a ticket and he/she belongs to a particular group. I can't find a direct association.
So this report basically takes a count of all the incident tickets where the user is the 'submitter' and displays the groups that he/she is associated with.
Note that you will have trouble because a user may belong to more than one group. So for example what might happen is UserX is a member of both groups A and B. The report will show that both groups A and B have a ticket, but it is referring to the same ticket.
If you have users in only one group then you don't have to worry.
Give it a go anyway, you might find this helps as a starter. I've made a couple of comments in the SQL to help you change the values if needed.