He said: "I was experiencing pain in my knee every time I fell over and I was having it heavily strapped.
"According to the surgeon Gordon Mackay, I could have left it until the end of the season, but then there was the probability the hole could have got bigger and resulted in me being out for three to four months.
"So this way we sorted the problem out early enough. It was more of a tidy-up operation than anything else. I think the general timescale for something like this is three to four weeks.
"I spoke to the surgeon, Gordon, and he said it went very well. It was the sort of operation that we weren't sure needed to be done or we could leave it. But he was very pleased that we did it."
Callam made just one appearance in Scotland's World Cup campaign, coming into the side for the Murrayfield clash with New Zealand, which saw the home side beaten 40-0.
He said: "The World Cup was an enjoyable experience for me, but I was just desperate to get some game-time, so I'm really looking forward to getting back to playing with Edinburgh."
Edinburgh centre Andrew Easson, 23, has also undergone ankle surgery and faces three months out of action.
Easson broke his left leg in April and said: "Hopefully all the bad luck is out of the window now."
Edinburgh coach Andy Robinson said of his recovering players: "We are looking forward to having them back 100% fit when they return."