The story is set during the American Civil War, so one external conflict that is present is the conflict that is raging between the Union and the Confederacy. The war was over many things, but the two main issues were states' rights and slavery.
Another external conflict is the conflict between Peyton and the Union soldiers that are going to hang him. Peyton was caught trying to sabotage a bridge, and his punishment is death by hanging. Peyton's escape could be viewed as an external conflict. He is struggling to not drown, get shot, etc. Of course that is all happening in his imagination, so maybe it is an internal conflict too.
A better choice for internal conflict is the conflict going on inside of Peyton before he gets captured. The story tells readers that Peyton really wanted to take a more active part in the war, but he was prevented from doing so for some reason.
Circumstances of an imperious nature, which it is unnecessary to relate here, had prevented him from taking service with that gallant army which had fought the disastrous campaigns ending with the fall of Corinth, and he chafed under the inglorious restraint, longing for the release of his energies, the larger life of the soldier, the opportunity for distinction.
Now he has been given the chance to earn some distinction for himself, but he must wrestle with the risk. If he fails, he will die and leave his family without a provider.
Another internal conflict occurs in the beginning of the story. Peyton is about to be dropped to his death, but he isn't panicking, nor is he begging for his life. He could be doing either, but he attempts to maintain a proud southern outward show of bravery. Even seconds before he dies, he is still thinking of escape.
"If I could free my hands," he thought, "I might throw off the noose and spring into the stream."