The basic premise of the question is to trace how Montag evolves as a character throughout Bradbury’s work. What events, moments, characters allow Montag to start the process of change that not only defines his character, but helps to form the arc of the work. I would say that the answer to this question probably starts with Clarisse asking if Montag “is happy.” I think that in trying to trace what allows Montag to question the rules of his society, one has to examine the interaction with Clarisse because it helps to develop a sense of self consciousness. Prior to this interaction, one could argue that there was something percolating, brewing, underneath the surface, but he never acted on it. With Clarisse’s question, the process of questioning the rules of his society begins as does his evolution of character.
You're probably having trouble understanding the question because it is so ambiguous. The question doesn't specify whether the question concerns the personal, the social, or the legal aspects of the issue.
For the most part, nothing allows Montag to question his society. He's not supposed to. His wife tries to deny him this right, her friends rebel when he reads poetry to them, and the law certainly doesn't allow him to question society. He gets his house raided once the authorities are tipped off that he has books in his house.
In short, then, the answer is that nothing allows Montag to question society.
That said, he does it anyway. Again, however, it's not because he is allowed to. He isn't allowed to.
No dictatorial society can control even a person's thoughts, however. Montag can think what he wants, as long as no one else finds out. He questions his existence and his society until the authorities find out. Then, of course, he is forced to flee.
Your teacher might possibly be looking for you to say that the fact that Montag is a firemen gives him a little bit of leeway. He does get to talk about books a little bit with his captain, and Beatty gives Montag a chance to return a stolen book--something a "civilian" might not get a chance to do. And Beatty at least suspects Montag of questioning society when he is motivated to send the mechanical hound to Montag's house in order to spook Montag into compliance. You might be able to argue that Montag gets breaks others don't. Thus, I suppose it is possible that that is the answer your teacher is looking for.
For the most part, though, nothing allows Montag to question society, unless it's just the fact that he is a human being capable of thought. He just does it anyway.
Clarisse moves him to think. The woman who martyrs himself impresses him. And Faber gets him thinking, too. These may enable Montag to question society, but I don't know if you could phrase it that they allow him to question society.
What this question is essentially asking you is "why is Montag able to be a rebel -- why can he question the way that the society is?" To me, the real reason why he can do this is that the society has not yet stamped out human nature. Because Montag remains human, he is able to question the society.
In this book, it seems that the society is trying to stamp out the parts of human nature having to do with thinking and with true human emotions (like love). This is why they have banned books and it is why they try to have people bombarded by entertainment all the time.
But the society cannot force all the people to stop wanting to learn and to have real relationships with others. Because the society cannot do this, people like Montag (and Faber and Clarisse and the old lady) are still able to question the society's values and rules.