How do you know that Jem doesn't believe Mr. Nathan Radley's explanation of why he filled the tree hole with cement in To Kill a Mockingbird? ...

The correct answer is He asks Atticus if he thinks the tree is dead.

During the summer with the influence of Dill, the children have become fascinated with the "haint" that lives across the street from them. Dill has suggested that Jem touch the house, and they also contemplate how they can communicate with the strange recluse who dwells within the walls of the Radley house.

Because the children have been within Boo's viewing, he...

The correct answer is He asks Atticus if he thinks the tree is dead.

During the summer with the influence of Dill, the children have become fascinated with the "haint" that lives across the street from them. Dill has suggested that Jem touch the house, and they also contemplate how they can communicate with the strange recluse who dwells within the walls of the Radley house.

Because the children have been within Boo's viewing, he has obviously watched them and, apparently, delighted in their play. One day as they pass the Radley tree on their way home, Jem discovers two soap figures that bear an incredible resemblance to himself and Scout; impressed at Boo's skill, he puts them in his trunk when he arrives home. On subsequent days they find chewing gum, a pocket watch, a medal, and other small items. So, Jem decides that they must write a thank you note to Boo; consequently, they leave one in the hole. But, the next day as they approach the tree, anticipating something new in it, they discover that the knothole has been filled with cement. Jem asks Mr. Nathan if he has filled it; he replies that he has, in fact, done so.

“Why’d you do it, sir?”
“Tree’s dying. You plug ‘em with cement when they’re sick. You ought to know that, Jem.”

Jem says nothing until the children meet Atticus as he walks home from his office. Jem asks his father,

“Is that tree dyin‘?”
“Why no, son, I don’t think so. Look at the leaves, they’re all green and full, no brown patches anywhere.

The answer from his father confirms Jem's doubts about the truth of what Mr. Nathan Radley has told him. Obviously, Mr. Nathan does not want Boo to interact with the Finch children.