A DoQueryCount wouldn't count records quite like that, it is instead set to count records that meet the nature of a query. For that format you might instead want to look at just DoQuery. It allows you to put in a query string and dictate how you want it to be returned. Specifically you can use the section on clist to dictate how you want that information to be returned. In it you could ask for the Parent Record to be displayed as the most useful fields to help you identify them and the # of related child records field.
For example if you had a Companies table and Contacts as their child records you could set it up to return the Record ID field, Company name field, and # of Related Contacts field as the result with a query to only show records that have Related Contacts. So you would get something like
7, Paper Co., 5
To say that Company 7, Paper Co., has 5 Related Contacts. That syntax is covered in the entry for DoQuery.
https://help.quickbase.com/api-guide/index.html#do_query.htmlAlternatively you can create a report in Quick Base that does much the same thing if you are just looking to get this type of information. The you can even set that report to email out on a schedule to specific users in the application. If you aren't needing just an API call.