Name the doctors that one can visit to be tested for menopause?

Any. Family practice, internal medicine, gynecology, medical endocrinology all should be able to test. Whether a particular doctor has interest or knowledge in this area are different among individuals.
FSH Test. It is best to find an integrative medicine physician who specializes in menopause. They are best trained to treat menopause appropriately with cyclic, bioidentical hrt. However, any doctor can diagnose it based on your symptoms and order an fsh (follicle stimulating hormone) test for you. Fsh >25=menopause.
There is no test . Menopause is a clinical diagnosis and usually does not require any testing it can be diagnosed by your family physician or gynecologist.So the answer to your question is family physician internist gynecologist.