There's no single, universally required exam for exterminators. Licensing and certification requirements vary significantly by state and sometimes even by county. Some areas might require a written exam covering pest identification, control methods, safety procedures, and relevant laws. Others might have apprenticeship programs leading to certification. Some might only require continuing education credits after an initial licensing period.
To find out the specific exam requirements (if any), you need to contact the pest control licensing board or department of agriculture in the specific state or area where the exterminator works or intends to work.