To unlock the Trust Me, I'm a Doctor trophy in Assassin's Creed Odyssey, you will need to wait until you reach Sequence 7 of the main story campaign.
Once at sequence 7, travel to north Thebes, Cursed Land of Oedipous, Boeotia and look for the side quest marker. When you have spoken to Hippokrates you can begin the following quests:
Let My Patients Go
When both of these quests have been completed, travel to south west Thebes and the next side quest makers will appear for the last two quests in this questline:
Too Much of a Good Thing