Although his Academy Award nominated appearance in The Russians Are Coming the Russians Are Coming (1966) is often cited as his debut film, Alan Arkin appears here uncredited as lead singer for the Tarriers. Since he was originally a singer, this is not an acting debut but his band on film. So the other movie is indeed his acting debut as he's playing a character.
The same year, Johnny Desmond and Merry Anders did two films together, the noir action Escape from San Quentin and the music exploitation Calypso Heat Wave.