What are some key quotes in the story that pinpoint the conflicting situation between Edgar, his mother, and claude?
There are several quotes that would help you pinpoint the conflict between these three characters, and rather than providing a list, it would be probably be more beneficial for you to look again at a couple of key chapters. Rereading these will help you decide which quotes are the best indicators of the conflict between Edgar, his mother, and Claude.
I think the real conflict begins after Edgar has seen his father's ghost in the barn, and knows Claude is a murderer. The chapter entitled "Smoke" from Part 3, "What Hands Do," is the when Edgar begins to withdraw from his mother and question her grief over the loss of his father. She senses his anger and says, "Don't take an attitude...What's bothering you?" To me, this quote defines the rest of the story, as no one, including Edgar, can define what is "bothering" him. Also, this is the night where Trudy tells Edgar that Claude will be coming to stay for a while, a fact which further separates Edgar from his mother. The brief, sort of climax to this section comes in the final chapter of Part 3 ("The Texan") when Edgar accidentally kills Dr. Papineau. His mother tells Edgar to run, but says she'll stand by the silo when it is safe to come back. This secret between her and her son is moment where she reveals that she might not trust Claude as much as Edgar believed she did. When Edgar makes the decision not to come back at all, however, he draws the final line between himself and his mother:
Then he stood and turned from everything he knew and the four of them [Edgar and the dogs] began to make their way into the dark Chequamegon (328).
The next section you should re-read comes on Edgar's return to the farm. The separation between himself, his mother, and Claude is never fully resolved as the story ends tragically and somewhat suddenly. However, the confusion, miscommunication, secrets, and ultimately, the separation is most clearly revealed in the barn burning scene in the final chapters of Part 5. Edgar runs in to save the records of the dogs. Trudy is pinned down by a blinded Glen Papineau shocked and horrified at the scene before her. Claude stands idly by, wondering what he should do, but making no haste to do anything. In three consecutive chapters, "Trudy," "Edgar," and "Claude," the same set of circumstances are revealed through each characters eyes, and it is clear that none of them are thinking the same thoughts as one another.