Alonso, a two-time world champion, was unhappy following his fifth-place finish, which saw him slip to third in the world championship standings and 39 points behind triple world champion Sebastian Vettel.
Alonso said that Ferrari "need to improve the car if we are going to have chances to win the world championship".
And while Montezemolo agreed he wasn't satisfied with the car's performance, he reminded Alonso that he needed to be more of a team player.
"All the great champions who have driven for Ferrari have always been asked to put the interests of the team above their own," said Montezemolo on the Ferrari website.
"This is the moment to stay calm, avoid polemics and show humility and determination in making one's own contribution, standing alongside the team and its people both at the track and outside it."
The article carrying the president's quotes expressly stated they were directed at Alonso, who turned 32 on Monday.
Rumours run wild
It said: "That was a reference to the latest comments from Fernando Alonso, which did not go down well with Montezemolo, nor with anyone in the team. So, when Montezemolo called the Spaniard this morning to wish him a happy birthday, he also tweaked his ear."
Alonso has recently been linked with a move to Red Bull, whose Australian driver Mark Webber is retiring at the end of the season.
His manager was alleged to have met with Red Bull boss Christian Horner in Hungary, although Alonso claimed not to have been aware of it.
Alonso said the four-week summer break in Formula One was a time when rumours ran wild.
"Always with four weeks with no Formula One it's a very dangerous period for us, and the media, but hopefully we will have a quiet summer," he had said Sunday.
Ferrari have failed to be consistently competitive since Alonso joined the team in 2010.
And yet the Spaniard has out-performed his teammate Felipe Massa and driven brilliantly to finish in the top four in each of his four seasons with the Italians, finishing runner-up on two occasions. – AFP