An external conflict is a problem outside yourself.
The most obvious external conflict is between society --run by the government - and the book lovers. Books have been deemed dangerous because they make people unhappy and discontent. Therefore all books must be destroyed. People report on other people in the society so that there will be no disruption of their lives. The citizens cannot do as they please, they must do what the society deems necessary for harmony or be destroyed.
Another external conflict is between Montag and Beatty. Beatty is Montag's boss and the head of the fire department. He tries to convince Montag that books can support ideas on both sides of an issue and will only create chaos. He is such a strong influence that Montag asks Faber for help in dealing with his ideas.
"He's read enough so he has all the answers, or seems to have. His voice is like butter. I'm afraid he'll talk me back the way I was." (pg 89)
When Beatty tries to arrest Montag and issues a warning to Faber that he will find him, Montag kills him, the ultimate external conflict.
A third conflict is with the Mechanical Hound. Montag senses that the Mechanical Hound doesn't like him from the beginning.
"Montag backed up. The Hound took a step from its kennel. Montag grabbed the brass pole with one hand. The pole, reacting, slid upward and took him through the ceiling, quietly......He was trembling and his face was green-white. Below the Hound had sunk back down upon its eight incredible insect legs and was humming to itself again..." (pg 26)
When Montag kills Beatty, the Mechanical Hound attacks him, and he burns it, destroying it. However, there is another one in another district. They bring that one and it hunts until it finds a victim and kills him.
The pages numbers I have given are for my book, but you should be able to find them somewhere close in your version.