The summer signing from Charlton Athletic, 26, suffered the injury during a training session on Friday.
"He had a nasty break of his finger on Friday which needed an operation. He'll be out for a few weeks," Pearson told the club's official website.
"It was pretty straightforward in terms of the surgery and he'll probably have another operation again this week and then we'll see how he is. The first prognosis would be that it will be about three or four weeks."
Hamer has made two appearances in all competitions for Leicester so far this season.