De Bruyne struck for Napoli during defeat at AC Milan, but was also substituted by Conte - and the midfielder made clear his unhappiness about being taken off.
Quagliarella told Sky Italia this morning: "Knowing Antonio Conte, he'll definitely send a written communication and ask De Bruyne during the week about his reaction after the substitution.
"Of course! I say this because it happened to me when he was coaching. I once made a joke while leaving the pitch after he'd made a substitution. His assistant Angelo Alessio was there and I said, 'You're mixing up these numbers a bit, eh? Why is the number 27 always used and I'm always used?'
"Then he said to me, 'What did you say?' And I said I'd told him to go to hell. I couldn't tell him I'd made a joke, seeing as we'd lost... He then insulted me in front of the whole team and I apologised."
Wrong person
Conte said of De Bruyne's in his post-match media conference: "De Bruyne? I hope he was upset with the result, because if he was upset with other things, he found the wrong person."