"Doc" Brown was the Paris family doctor during the 2350s.
He always had lollipops in the waiting room, no holocomics more than six months old, and even made house calls. When Tom was nine years old, he caught a bad cold; the doc whipped up a pot of garlic soup and brought it over himself.
In 2371, while undergoing scans by The Doctor, Paris waxed nostalgic about "good old Doc Brown", implying he'd rather be under Brown's care than The Doctor's. The Doctor, for his part, found Paris's reminiscences "irritating". (VOY: "Cathexis")
This character was only mentioned in dialogue.
The Star Trek Encyclopedia (4th ed., vol. 1, p. 108) identified Doc Brown as a pediatrician.