I have a patient who is a doctor and I am confident that s/he is clinically depressed, but they deny this saying they are worn out from work. How should I manage this case?

Doctors have a reputation for not recognising symptoms of illness in themselves. They deny their own symptoms and delay seeking medical advice for themselves. Work comes first. I think you are in a god position because this doctor is attending you for something, advice or screening and that is half way there. On a future encounter I would suggest using a validated depression assessment tool like the PHQ-9 questionnaire and this can help to demonstrate objectively to patients who are medically qualified that they have a mood disorder that needs to be addressed.