Davies has started seven of Blues' 13 games this season in the league in which they sit seventh, 11 points off the play-off places.
And, while the surgery rules Davies out of the entire RBS 6 Nations, Davies hopes to see the 6ft 6in forward back in time for the business end of the season.
"Bradley had the surgery on Thursday which went well and the surgeons have done an excellent job," Davies told Wales Online. "By Monday we will know more about when he is back.
"It is going to be a few months but whether it is two or three we are not sure at the moment. But we are expecting to see him back before the end of the season."