A return date for the 32-year-old has not been set, with the club stating that they are hoping that Rees will come back into the team in the "future".
A statement on the club's official website said: "Cardiff Blues have today announced that club captain Matthew Rees is to have an enforced spell away from rugby.
"Cardiff Blues would like to take this opportunity to wish Matthew a speedy recovery and we look forward to seeing him back in action in the future.
"The region will be making no further comment and would ask that the media respect Matthew's privacy at this time."
Rees, who has earned 58 Wales caps, joined the region from Scarlets in the summer.