The England international watched on from the stands at the San Siro to witness his new side record a thrilling injury-time comeback win over Parma on Sunday.
Milan scored two goals in stoppage time courtesy of Tijjani Reijnders and Samuel Chukwueze to emerge as 3-2 victors to move up to sixth spot in Serie A.
It was Conceicao's fifth win from his seven games in charge of the Rossoneri, but not even his delight at a last-gasp winner could spare Calabria from his rage.
In shocking scenes, the ex-Porto manager was spotted confronting the Milan right-back on the pitch and pointing in his face.
A verbal spat then ensued where the duo shouted at each other, leading to players and staff members having to get involved.
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Conceicao and Calabria had to be dragged away from each other with fans inside the stadium left stunned by what they were witnessing.
It remains unclear what sparked the altercation, though La Gazzetta dello Sport have claimed that Conceicao wasn't best pleased with his captain attending the concert of rapper Lazzo on Friday night.
Calabria, who was withdrawn in the 77th minute when the score was level at 1-1, has since revealed that he and Conceicao have patched up their differences.
He said: "Honestly, these are things that happened on the pitch. A misunderstanding between the two of us, we wanted to play this game.
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"The adrenaline was really high and we cleared things up, we both didn't understand one thing.
"We fixed things: it won't be the first or the last time you see it. It's common in football. I also apologise because it's not a nice thing.
"The most important thing is the team and having turned the game around."
Conceicao also discussed the incident, insisting his issue with Calabria was due to something he did in the game.
He remarked: "I watch the matches and feel them with great passion. There was a bit more adrenaline.
"There was a game situation that I discussed with Calabria, and there were a few words too many, in my opinion.
"The players are all important to me, like a family. If I have a son who behaves badly, I tell him. And here, it's the same.
"There's no hypocrisy: we say things to each other as men. True, it wasn't pretty, but in football, it happens, we're not in church.
"We clarified things. Sorry for the delay, but I was already planning tomorrow's work."
While Calabria found himself in Conceicao's bad books, he may also soon find himself out of his starting XI.
He seems destined for a place on the bench following the loan arrival of Walker from Manchester City.
But the 34-year-old will have to wait until February to make his debut due to a little-known rule.