You can find this incident towards the end of Part 1.
At first, Montag goes into the house on Elm St. as a committed fireman. He goes in intending to destroy the books and the house. He gets annoyed with the old woman because she is not playing along -- he wants to do things the right way. This shows that he is still a normal fireman.
But then something happens. The woman starts to talk about how she is not going to leave and she implies she's going to die with her books. This bothers Montag and he starts trying to persuade her to leave. Clearly, his thinking has changed.
Later, he is talking to Millie about the woman. He is disturbed by what has happned and he has started to wonder if there must be something important about books -- otherwise, why would that woman die for them?
So the interaction with the woman really changes his thinking about her and about his whole society.