As manager of the away team, Mourinho had expected to be allowed to address the media first so that his team could set off on their journey back to West London, reports said.
"Roberto, next time tell me go before you because we have to travel," Mourinho was quoted as saying in the reports.
"We don't control that, Jose. I don't control that," Martinez said in reply.
To which Mourinho responded with an expletive.
"When he beat us 6-3 (last season), he was such a nice man. I prefer him like that," Martinez said after the Chelsea boss stormed away.
The loss to Everton, courtesy of a Steven Naismith hat-trick, was Chelsea's third defeat in five Premier League games and they now sit fourth from bottom in the table.
The relationship between the clubs was already strained after Chelsea's public pursuit of defender John Stones, who had his transfer request rejected by the Merseyside club.
Chelsea will host London rivals Arsenal, who beat Stoke City 2-0 during the weekend, on Saturday.
(Reporting by Shravanth Vijayakumar in Bengaluru; Editing by Sudipto Ganguly)
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