Several things happen during the novel that show that Jem is growing up. He takes less interest in his sister, childhood games and trying to make Boo Radley come out. He spends more time on his own, reading and Scout notices he's become moody and is eating more--all signs that Jem is, physically, growing up and going through puberty.
When Nathan Radley fills the knot hole of the tree with cement, it is Jem who realizes how cruel Nathan is being to Boo, and it is Jem who cries about. Jem also stands up to his father in front of the jail house and tries to protect his father. Jem is broken-hearted about the verdict of Tom Robinson's trial, but is able to have a mature, logical discussion with Atticus about what happened and what the outcome might be. Jem is the one who make the decision to tell Atticus that Dill has run away, rather than turning it into a game or hiding him from Atticus. Jem also tries to fend off Bob Ewell and protect Scout at the end of the novel.
These actions on Jem's part, rather than being more concerned about not getting in trouble with Atticus or making Boo Radley come out of his house, show that Jem is growing up and maturing.