In Pixar's Cars 3, Doc Hudson won his first Piston Cup in 1951.
Doc Hudson, also known as the Fabulous Hudson Hornet, is presented as a legendary former racer. During his racing career, he won three Piston Cups and became one of the most respected figures in the history of the fictional competition. Cars 3 uses this history to connect Doc's experience with Lightning McQueen's struggle against a younger generation of racers.
The film reveals that Doc's first Piston Cup victory came in 1951. He then won again in 1952 and 1953, establishing a three-year championship streak. His career ended after a crash and an injury, after which he became Radiator Springs' doctor and later served as Lightning's mentor.
The year 1951 is specifically the beginning of Doc's championship run, not the year of his final Piston Cup victory. The three consecutive seasons—1951, 1952, and 1953—are often remembered together, so the question's wording identifies the first one.