Martinez was alleged to have told a Spanish newspaper that Allardyce viewed himself as Ferguson's successor at Old Trafford and was part of a group of managers sympathetic towards the Manchester United boss.
Martinez subsequently denied making the comments and Allardyce confirmed that the Spaniard had phoned him to apologise.
Allardyce, whose team take on United at Old Trafford in the Premier League on Saturday, accepted Martinez's apology but warned him to be more careful in future.
"I accept what he said when he rang me" Allardyce said. "When I saw the papers I thought, 'Is it true or isn't it? But I have to take his word for it, which I do.
"Welcome to the Premier League, Roberto. Keep your mouth shut in the future or make sure you remember what is going to be done in the Premier League.
"If they have distorted his words, then that is something of a regular occurrence at this level.
"It's not a real problem for me, I just think Roberto has learnt a harsh lesson about what happens in the Premier League if you start diverting away from you and your club business.
"You can say you can be misinterpreted, you can say you can be read wrong, but it's a lesson that when you comment in this league on anything it can be blown out of all proportion.
"He's apologised to me and I accept that apology."
Blackburn go into Saturday's match having failed to pick up a point in their last nine away league games and Allardyce has called for a greater mental strength from his team.
"It's about the mentality of the players and we have to have the same desire to compete against the opposition away from home as we do at home," he said.
"We just seem to stand off the opposition, whereas at home we are all over them like a rash.
"It is a big concern for me, the staff and the players."
Jason Roberts is unlikely to be available for selection after becoming the latest Blackburn player to show symptoms of swine flu.
Allardyce, who has suffered with the illness himself in the last few days, said that he and most of his squad had recovered after an outbreak of the virus in the camp last week, but confirmed that Roberts was being seen by a doctor.
"I'm okay now, I think we've only Jason Roberts," Allardyce said.
"There was another player suffering with it earlier in the week and Jason looks like it now, so we've left him at home and sent the doctor out to see him."
David Dunn, who was thought to have suffered with the virus, looked in good health when he came off the bench against Peterborough in the Carling Cup on Tuesday and is expected to play tomorrow, while another victim, Chris Samba, is back in training but still feeling weak.
"We've just got Chris Samba back, who seems to have been affected the most by the symptoms," Allardyce said.
"He's been left pretty weak, I think he has lost about four or five kilos. It is an incredible amount of fluid he has lost.
"We'll have to check if he is strong enough to be capable of being selected for the squad on Saturday.
"Pascal Chimbonda is fit, Vince Grella is back in training but it might be a bit too soon for him. The squad has strengthened from last week, which is very important when you are going to Old Trafford."