The Scottish Premiership champions signed the 20-year-old striker for £9m last week following his successful loan spell at Parkhead from Paris Saint-Germain last term.
Edouard scored 11 goals in 29 appearances in his debut season in Scotland as the club secured a historic back-to-back domestic treble, and Brown is confident that his new teammate will continue in the same vein despite his price tag.
"He'll not be bothered," Brown told Sky Sports News. "He is very laid back so that's a great positive. He does his talking on the park and that's where he needs to do it.
"He will develop a lot under the great coaching team that's here. I think that's what the manager sees in him. He is young and he's a great prospect as well. That's kind of the way football is going as well, we need the pace and qualities of youth. Especially when you have got me in the team.
"He's a great lad in the dressing room and he showed his qualities on the park. He shows it in training every day and he showed it on the big stage as well. It's great he has come and great that he wanted to be here after last season as well. I am delighted he came back in."
Edouard spent seven years in PSG's system without making a single senior appearance.