I say no. Ok It's been a long time since microbiology. Maybe someone with more recent college can correct me but I thought it worked on only gram negative bacteria or maybe just gram positive. Either way it's not broad spectrum and not even used in the medical community for much anymore.
for sterilization you should be using at least 70% isopropyl alcohol. peroxide may work for cleaning wounds, but i don't think it works for implements.
For quick cleaning, just use a brush cleaner designed for mu brushes. At least twice a month clean your brushes with a gentle shampoo and conditioner (conditioner not needed for the synthetic brushes).
Wouldn't hydrogen peroxide change the color of the hair in the brush? I mean it does to human hair?? It would be too harsh anyway. I always use shampoo, and it works great!!!