The central conflict in Fahrenheit 451 is Montag's decision to rebel (or not rebel) against the rules, laws, and values of his society. This is an internal conflict.
The novel goes a long way to define exactly what these rules, laws and values are.
Free thinking is essentially outlawed in favor of a legally mandated group mentality. Montag realizes that he has a choice to make. He realizes that he can choose to think for himself.
This option is symbolized by the books that Montag hides in his home. The books represent both Montag's rebellion and a choice to think.
Once Montag makes his decision to rebel against the group mentality of his peers, the internal conflict which drives the novel becomes external. Montag commits a crime and becomes an outlaw, a free-thinker, and he joins a small group of similar people who have chosen to read and to think for themselves.
(To be clear, the main conflict takes place over the entire novel and shouldn't be confused with the novel's climax.)