Not exactly. Custom fields can't be restricted to particular user groups directly—however, you can associate custom fields with particular departments, and then restrict those departments to specific user groups to achieve the same effect.
To do this, start by creating custom fields from Admin CP > Custom Fields > Groups > New and link them with associated departments.
Further, the same department can be assigned to only registered user group. It can be achieved from: Admin CP > Departments > Manage Departments > Permissions : Users > Restrict to Custom User Groups = Registered
As a result, when any of the registered user will log in at the Support Center, they will be able to see the above departments only when submitting tickets, and the associated custom fields will be shown upon selecting the department.