As long as he is flurrying, yes.Otherwise, no.
They don't need to be flurrying for this.
Unarmed Strike: At 1st level, a brawler gains Improved Unarmed Strike as a bonus feat. A brawler may attack with fists, elbows, knees, and feet. This means that a brawler may make unarmed strikes with her hands full. A brawler applies her full Strength modifier (not half ) on damage rolls for all her unarmed strikes.
Flurry doesn't even come in til 2nd level. It lets them apply their full Str mod with any brawler weapon, not just UAS.
hmm interesting. I thought maybe players did half damage with unarmed strikes but they don't. This appears to mean if you make 2 attacks even if your BAB is 5, you take the two-weapon fighting penalties to hit, but get full STR mod added to your offhand. that is nice. That would mean a brawler actually gets another "free" feat i didn't know about over another character that takes Improved Unarmed Strike.