The Posh defender, who joined The Canaries on loan at the end of November, has moved to Carrow Road for an undisclosed fee.
"I stated my desire to stay here permanently as soon as I saw the size of the club, the fanbase, the stadium and the facilities at the training ground," Martin told the official Norwich website.
"Thankfully I've done enough to prove to (manager) Paul Lambert in the four weeks that I'm worthy of staying and I'm really enjoying playing for him as I think most of the lads do."