The 26-year-old claims that he was threatened by the coach and trainer of his doubles opponent following Friday's match.
Kyrgios and Kokkinakis knocked out top seeds Nikola Mektic and Mete Pavic in just the second round of the men's doubles tournament.
A raucous crowd helped the Aussies to a 7-6 6-3 victory, however the flash point of the match came when Kyrgios hit one of his opponents with the ball.
The player took to social media after the match saying: "Just letting you know after yesterday's chop fest in doubles my opponents coach and trainer proceeded to threaten to fight in the players gym.
"Tennis is a soft soft sport @TKokkinakis all because I moved and hit them with a tennis ball."
Kokkinakis replied: "That was crazy!! Mans thought it was @ufc."
Mektic and Pavic were the pre-tournament favourites but found themselves in the firing line for their Australian opponents and the capacity crowd.
Earlier at the Australian Open, Daniil Medvedev was critical of the crowd after his singles victory over Kyrgios, and Pavic said: "They are pretty loud. Obviously they cheer for Aussies. It wouldn't hurt them to show some respect to all opponents, to other players."
Mektic added: "It all went very fast, very loud. They're a nasty team to beat here and that's it."