The 27-year-old suffered the injury in training on Tuesday and will miss the start of the new Championship season as a result.
Davis tweeted that he hopes to come back stronger from injury, which could keep him out until the autumn at least.
He said: 'Op successful! Ill be back stronger than ever, thanks for the messages of support.'
Birmingham returned for pre-season training last week ahead of Garry Monk's first full campaign in charge.
They fly to Austria on Friday for a training camp and kick off the Championship season at home to Norwich on August 4.
Davis has made 167 appearances for Blues, scoring 11 times, since joining from Wolves in 2014.
Didier Deschamps praised France's performance after his side beat Croatia 4-2 in Sunday's World Cup final at the Luzhniki Stadium.
Toby Alderweireld believes Tottenham fans should be proud that the club had so many representatives at the World Cup.
Roy Hodgson says Crystal Palace are 'not a selling club' and have no intention of selling Andros Townsend or Wilfried Zaha.