You are right, this thing was never explained. DoC Q was already a member of the Blackbeard pirates, so Blackbeard was definitely not searching for a doctor crewmate. I could think of a couple of theories:
(1) Blackbeard was trying to gain more information about his abnormal physique from the medical experts of Drum Island.
(2) Blackbeard was originally from a winter island (based on Luffy's statement that the winter island people never sleep) and he took out his childhood sadness and anger on Drum, another winter island