The 25-year-old, the Dons' player of the year last season, was taken off on a stretcher during the first half of the 3-1 win at Dundee on Saturday.
"It was very upsetting to find that Andy had a broken leg after the game," Brown told RedTV. "It looked very serious at the time of the accident, and it was an accident. Sometimes the tackle has been violent or malicious, I'm most certain it wasn't that.
"Andy was very unfortunate and he went to Ninewells Hospital. Archie (Knox, the assistant manager) and I went in last night, but unfortunately he hadn't come back from the operating theatre."
Brown spoke to Considine on the phone on Sunday night and the operation was a success.
"Ryan Jack (who broke his leg in October) this morning (Monday) was starting to run and train with us," Brown added. "It will be a delight when I see that starting to happen in a few months with Andy Considine."
Considine is expected to miss most of the rest of the season.
Meanwhile, St Johnstone will have Peter Pawlett available for their home Clydesdale Bank Premier League game against Dundee on Wednesday after Aberdeen agreed to extend the midfielder's loan deal for another three weeks.
The arrangement allows Pawlett to continue with Saints for the matches against the Dens Park men and the first match after the short winter break - the Sunday fixture against Motherwell at Fir Park on January 20.
Manager Steve Lomas told St Johnstone's official website: "I'm grateful to Craig Brown for agreeing this. I can understand that he doesn't want to commit to anything beyond that for the moment and it will be good to have Peter available for the two games that this extension takes in."