Why is this story a quest for justice?
Perhaps the term "quest for justice" refers to the need for justice being served in regards to the guilty being punished. Although the narrator tries to convince us (and probably himself as well) that he is not mad and the murder was justified, his conscience will not let him get away with it. Although there were no witnesses and he had hidden the body under the floorboards and he might have gotten away with it, he continues to hear the beating of the victim's heart. It is telling the tale of a dastardly deed. He continues to hear the constant beating in his mind until he can no longer stand it and screams out his confession to the police and...justice is served.
When the police finally arrive at the end of the story, it becomes a quest for justice. Whether or not the narrator will be apprehended was probably never in doubt. The police will put the clues together and arrest the narrator, or the narrator's guilty conscience will give him away, and he will also be arrested under those conditions.
The narrator's action all throughout the story become more ironic as the ending becomes clearer. The narrator went to great lengths to commit the act of murder but didn't have the conscience to pull it off at the every end.
I don't know if I would use the term quest for justice to relate to the man's behavior. However, when people commit a crime they often subconsciously leave clues or engage in self-destructive behaviors that lead to their capture.
In the short story "The Tell Tale Heart" the man begins by telling his story. He needs someone to know what his act was and why he had done it. He tries to convince the reader that he had not been mad in his actions, but that killing the man was necessary because of the man's evil eye. The man tries to rationalize his actions which demonstrates that he is trying to prove that he was just in his actions of murder.