Nintendogs does not track a "naptime record." Unlike Obedience Trials, Agility, Disc Competitions, or Owner Info stats, the game logs no metric for how long, how often, or how quickly your puppy sleeps.
So while you can't earn a "Gold Napper" trophy, a well-rested pup is a happy, obedient, high-scoring pup. Let sleeping dogs lie—literally.