Montressor appears to be concerned about his companion's health in order to make Fortunato trust him. true or false

True.
In his introduction to his tale of revenge, Montresor states,

It must be understood that neither by word nor deed had I given Fortunato cause to doubt my good will. I continued, as was my wont, to smile in his face, and he did not perceive that my smile now was at the thought of his immolation.

Part of the success of this plan, then, depends upon Montresor's manipulation of Fortunato's ego."He had a weak point....He prided himself on his connoisseurship in wine." So, Montresor tempts his foe by telling him that he has just acquired a large cask of Amontillado. He feigns humility, saying that he has been foolish enough to pay the full price without consulting Fortunato first. But, because he could not locate Fortunato and seek his advice, Montresor went ahead and bought the cask, fearing that he might lose this opportunity. Now he would like Fortunato to taste it and confirm its value.
Once he has lured Fortunato into his family's catacombs, Montresor pretends that he is solicitous about his foe's health: "We will go back, you will be ill, and I cannot be responsible. Besides, there is Luchesi—" Not wanting anyone else to be the one who judges the wine, Fortunato insists that his cough is "a mere nothing" and adds, "I shall not die of a cough." When Fortunato makes a toast, "I drink to the buried that repose around us," Montresor adds, "And I to your long life." This continued pretense of concern for Fortunato's health keeps Montresor's victim from suspecting anything will happen to him.