There are a few quotes in the book that show Montag beginning to question his role as a fireman and a few, certainly towards the end, that show he has changed his views about burning books.
Montag really begins to question things after a few conversations with Clarisse McClellan. During one of those conversations, she noted that people used to put out fires rather than starting them. Montag denies that this ever occurred, but it evidently stuck in his mind. After his last conversation with Clarisse, Montag is at the firehouse and he is talking to Beatty and asks, "Didn't firemen prevent fires rather than stoke them up and get them going?" Just prior to uttering this line, Montag uses the phrase "once upon a time" and Beatty calls him on it. Montag worries that Beatty will become suspicious. (Montag had read the first line of a book of fairy tales at a previous fire.)
Just after this exchange, the firemen are called to a house and Montag steals a book. This shows his curiosity about books. More significantly, the owner chooses to stay in the house and burn with the books. Stealing a book and being so traumatized as this woman burned with her books is when he really realizes that burning books may be wrong. Montag tries to get Mildred to question things as he does. During this conversation, he tries to convey the passion of the woman in the fire:
"You weren't there, you didn't see," he said. "There must be something in books, things we can't imagine, to make a woman stay in a burning house; there must be something there. You don't stay for nothing."
This is the first definitive moment that Montag speaks aloud that there must be something important about books and therefore, it might be wrong to burn them.