A Quarterly physical is a complete medical work-up performed by a ships medical staff about 4 times a year.
During the 23rd century, doctors on Federation starships had the duty of performing quarterly physicals on each member of the crew. (TAS: "Yesteryear")
By the 24th century there was a different practice, of annual physicals done at staggered intervals. (TNG: "Allegiance")
- Episodes in which regular physical examinations are known for occurring are: "The Corbomite Maneuver", "Amok Time", one week before "Journey to Babel" and "For the World is Hollow and I Have Touched the Sky" (TOS). "The Corbomite Maneuver" episode establishes that these were indeed quarterly, and even though the physicals were never called quarterly again, they never conflict with the idea. "For the World is Hollow and I Have Touched the Sky" establishes that physicals had been done on the entire crew within a week's time, but didn't specify if they were quarterlies.