Yes, doctors should go through mental training courses to stop suicide rates from rising, both among doctors themselves and their patients. Being a doctor is a high stress job that can unfortunately end in suicide, and doctors surely must deal with many suicidal patients. Both may be curbed with more training.
Should surgeons who never deal with conscious patients go through training for this? No. Should psychiatrists and psychologists go through this training? No. They should already know this. Should pediatricians and other primary care physicians go through this training? Yes. Because they may not have enough training in this field, and they are more likely to encounter people who need help in this area.