The incident took place with 12 seconds remaining in the first quarter of Boston's 130-120 home triumph over the Detroit Pistons.
Brown made a throat-slash gesture after a dunk over Detroit's Isaiah Stewart.
Brown, in his ninth NBA campaign, led the reigning NBA champion Celtics against the Pistons with 28 points, nine assists, six rebounds and three steals.
At 18-4, the Celtics have the second-best record in the NBA, just behind Eastern Conference leader Cleveland's 20-3 mark, and have won nine of their past 10 starts.
Brown, 28, is averaging 25.2 points, 6.4 rebounds and 4.6 assists a game for the Celtics.
