Bonucci sensationally left Juve in the off-season, the Italy international centre-back joining Milan in a €42 million deal.
The 30-year-old's departure from Turin, shortly after Juve lost to Real Madrid in the Champions League final, sparked speculation of a row with Allegri. Those claims, however, were dismissed by Bonucci's wife, Martina Maccari.
"His decision wasn't due to an argument with Max Allegri," Maccari told Rai Radio 1.
"In a professional setting, I think balance between all parties is fundamental. Leonardo chose to leave because of the vicissitudes within the environment, but now he's calm and giving his all for this new adventure."
Bonucci's switch to San Siro has reinvigorated Milan — a sleeping giant in Italian football following years of mediocrity.
The veteran defender has been handed the captain's armband and great responsibility at Milan, who have not won the Scudetto since 2010-2011.
"I won't hide it, I'm not used to it yet and maybe he isn't either. He chose to come to Milan completely autonomously though, I just stood by him as a good wife has to do," she added.
Milan have won three of their opening four matches this season to sit fifth in the Serie A standings.