I think that one conflict that exists in the story is the conflict between Arnold and his parents upon receiving the news that Eugie is dead. The family is in conflict because they are unaware of why the death has happened. Arnold is in conflict because he recognizes that he is an outsider to the family. Steeped in their grief, Arnold is torn between wanting some type of validation, but also not wanting to intensify the already volatile atmosphere. His hiding in the barn is one way this conflict reveals itself:
Arnold went into the barn, down the foddering passage past the cows waiting to be milked, and climbed into the loft. After a few minutes he heard a terrifying sound coming toward the house. His parents and Nora were returning from the willows, and sounds sharp as knives were rising from his mother's breast and carrying over the sloping fields. In a short while he heard his father go down the back steps, slam the car door and drive away.
The conflict between Arnold and his parents upon hearing the news of his older brother's death is reflective of the collision between both sets of forces. Arnold's hiding in the barn is a part of this conflict.
Another source of conflict is an internal one in terms of how the Sheriff sees Arnold. When confronted with Arnold's lack of articulated remorse about what happened to Eugie, the Sheriff is conflicted about how he views Arnold:
The sheriff swung away from him, laid both hands flat on his desk. "Well, all I can say is," he said across to Arnold's father and Uncle Andy, "he's either a moron or he's so reasonable that he's way ahead of us." He gave a challenging snort. "It's come to my notice that the most reasonable guys are mean ones. They don't feel nothing."
Whether Arnold is "a moron" or a cold and detached killer is where the Sheriff's conflict lies. To a certain extent, it is the fundamental conflict that underscores the entire story. The Sheriff's conflict is the family's and our own because it is essential in being able to form some judgment about Arnold as a character and about his actions.
Another conflict that this precipitates is Arnold's need for some type of emotional validation. Arnold is in conflict when he goes to his mother's room at night. Herself in conflict, she rebukes him. When she asks him in the morning if he approached her last night, his response is one of conflict:
"I didn't want nothing," he said flatly.
Then he went out the door and down the back steps, his legs trembling from the fright his answer gave him.
The "trembling" of his legs is a reflection of the internal conflict that Arnold experiences. It is one where the basic question of why he did what he did forms the central conflict. In his response, it is evident that he has forged some conflict from within, moving on from his mother at a point where he needed her. In this light, conflict becomes one of the only certainties one can offer regarding Arnold.