No, John Davidson who appears in old movie credits has no relation to TV’s John Davidson.
The earlier John Davidson (1886-1968) was a character actor who usually played dapper, sinister foreigners in silent and sound films.
His movies included The Green Cloak (1915), Romeo and Juliet (1916), Dinner at Eight (1933), A Tale of Two Cities (1935), and Dick Tracy vs. Crime, Inc. (serial, 1941).
The later Davidson (1941–) is best known for TV shows like “The Girl With Something Extra” (NBC, 1973-74) and “That’s Incredible” (ABC, 1980-84).
At one time he had planned to enter the ministry.