Martinez's problem flared following Spain's friendly win over Ecuador in Guayaquil, and the European and Bundesliga champions decided on serious action this early in the season.
The 25-year-old is set to miss their UEFA Champions League opener at home to CSKA Moscow, along with league clashes against Hannover and Schalke.
Following the UEFA Super Cup win over Chelsea in which he scored a late equaliser, Martinez told reporters he was able to play for '20, 30 minutes' but not a full 90 due to pain.
Martinez's injury creates a dilemma in midfield for coach Pep Guardiola, as Thiago Alcantara, Mario Gotze, Bastian Schweinsteiger and now the Spain international are all unavailable for the immediate future.
The former Athletic Bilbao playmaker tweeted a picture of himself following the surgery, saying the procedure was a success.
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