"You see these hands? These are surgeon's hands, created by the most sophisticated computer imaging technology available. They do not play games, and they do not mop floors.
A surgeon was a medical doctor, such as a physician, dentist, or veterinarian, who specialized or was proficient in performing surgery on patients.
Pyong Ko was a 21st century surgeon who discovered the genetic sequence for inhibiting the growth of cancer cells. (VOY: "Fury")
Doctor Phlox once witnessed how a surgeon replaced the parietal lobe of a Bynar newborn with a synaptic processor. (ENT: "Regeneration")
In 2155, Earth Starfleet officer J. Billingsley worked as a surgeon at the Fleet Operations Center on Earth. (ENT: "Demons" set artwork)
Jadzia Dax, her symbiont Dax having met future Doctor Leonard McCoy at the University of Mississippi as Emony Dax, noted that she knew he would become a doctor because "he had the hands of a surgeon." (DS9: "Trials and Tribble-ations")
Doctor Leonard McCoy indeed became a qualified surgeon. (TOS: "The Man Trap", et al.) In 2268, he performed open heart surgery on Vulcan ambassador Sarek. (TOS: "Journey to Babel")
In the 23rd century, the chief medical officer of a starship was sometimes referred to as the ship's surgeon. (TOS; Star Trek III: The Search for Spock, Star Trek VI: The Undiscovered Country)
Doctor Katherine Pulaski was a qualified surgeon. In 2365, her expertise saved the life of Captain Jean-Luc Picard after complications arose during a procedure to replace his cardiac implant at Starbase 515. (TNG: "Samaritan Snare")
Doctor Beverly Crusher was a qualified surgeon. In 2368, she assisted Doctor Toby Russell in using a genetronic replicator to create a new spine for Lieutenant Worf, after he had an accident which paralyzed him. (TNG: "Ethics")
The original chief medical officer of the USS Voyager once worked as a surgeon in a hospital on Caldik Prime. (VOY: "Caretaker")
The Vidiians employed surgeons to remove organs from both living and dead aliens in order to transplant them into Vidiians to ensure their race's survival against the Phage. (VOY: "Faces", et al.)
An interrogation surgeon was a specialized position within Voth society used to question prisoners and those suspected of violating doctrine. In 2373, Haluk called for an interrogation surgeon to begin questioning Captain Kathryn Janeway, relating to Forra Gegen's defense of the Distant Origin Theory, but the surgeon was recalled when Gegen surrendered. (VOY: "Distant Origin")