Montag has both external and internal conflicts. His most important conflict is between himself and the government -- Man versus Society. He wants to develop his individuality through reading and learning, while society wants him to be informed and controlled through television. This struggle is initially internal, but becomes external when Beatty confronts him and calls him out on his crimes. Another conflict comes between Montag and his wife, Mildred, who is superficially happy but may have a deep, even suicidal misery hidden underneath:
"You took all the pills in your bottle last night."
"Oh, I wouldn't do that," she said, surprised.
"The bottle was empty."
"I wouldn't do a thing like that. Why would I do a thing like that?" she asked.
"Maybe you took two pills and forgot and took two more, and forgot again and took two more, and were so dopy you kept right on..."
(Bradbury, Fahrenheit 451, Google Books)
When Montag reveals his books to Mildred, she worries about their social standing; when they read together, she doesn't appreciate the books and their ideas. Instead, because she doesn't have the intellectual basis to understand satire, or philosophy, she falls back on television to comfort her. Montag's developing mind clashes with her atrophied mind, and he finds himself unable to relate to her anymore. This conflict also changes from internal to external: earlier, when Montag questions her television shows, he is internally conflicted; later, when he reads to her guests, he is expressing his dissatisfaction externally.