Captain Janeway is caught by surprise when she is paid a visit by Q and his son, Q ("Q2"). She last saw the young Q as an infant four years ago, but he already looks and acts like an adolescent, and his father wants to leave him on Voyager to learn about humanity. The crew quickly learns that the boy is trouble, being as bored and out-of-control as any teenager but with omnipotent powers. Q2 throws a party around the warp core, he makes Seven of Nine's clothes disappear, and he gives the computer a rebellious personality. Neelix is shocked to find him initiating a war between two cultures, and when he tries to talk him into using his powers more constructively, Q2 takes his vocal chords away. Then Q2 lures three Borg Cubes to attack Voyager, putting the crew in danger.
The father Q then shows up, sets things back to normal and speaks privately with Janeway. He tells her that his son was supposed to inspire peace and compassion, but instead has brought chaos to the universe, so the Continuum is hounding him to straighten the boy out. He was hoping that Janeway's "Starfleet ideals" would rub off on him, but she tells Q to act more like a parent and spend time with him. Q loves the idea, kisses her on the lips and disappears. Minutes later, Q appears again in Janeway's bubble bath, telling her he just spent years with his son smothering him with attention, and things have only gotten worse. Janeway tells Q that he needs to make his son understand that there are consequences to his actions. In response, Q turns his son into an Oprelian amoeba, and when he restores him, tells the boy he will be sentenced by the Continuum to an eternity as an amoeba unless he becomes an upstanding citizen of the cosmos. He gives Q2 one week to change his ways, and strips him of his powers.
Forced into playing "mommy" to the now-mortal boy, Janeway assigns Q2 quarters and devises a strict curriculum for him with crewmates as his instructors. For instance, Chakotay runs a diplomacy scenario on the holodeck where Q2 is instructed to settle a mining dispute among several races. But when Chakotay leaves the room, Q2 alters the aliens' personalities to assure speedy success. He is also assigned to write a historical essay on the Q Continuum, but Q2 manipulates Icheb into writing it for him. Janeway tells Q2 that she knows he's been cheating, and orders him to his room to wait for his father to return. Not wanting to live as an amoeba, Q2 explains that it's not easy to live up to his father's expectations and tells her she's his only hope. Janeway agrees to give him one more chance.
Janeway and the Voyager crew become impressed with Q2's determination to improve himself. Janeway is also pleased with Q2's new essay on the Q Continuum. She rewards Q2 by allowing him to go with Icheb on his piloting lesson with Lt. Paris. After helping Icheb navigate the Delta Flyer through an asteroid field, Paris lets Q2 take a "turn at the wheel." Q2's nervousness gives way to a sense of adventure, and he and Icheb become friends. Meanwhile, Q returns to Voyager to check up on his son's progress. Janeway has Q2 read his essay to his father, but Q is not at all impressed, hurting his son's feelings. When Q2 leaves the room, Q tells Janeway that the boy needs to demonstrate nothing less than exemplary "Q-ness." Janeway prods him to explain what that is, but he just says he knows it when he sees it, and an essay is not it, and disappears. Janeway visits Q2 to console him, telling him that if the Continuum won't take him back, she'll ask them to let him stay on Voyager as a human. But Q2 doesn't want to be human — he wants to be a Q, like his father.
Later, Q2 convinces Icheb to help him repair an ion imbalance on the Delta Flyer to surprise Paris. After making the repair, Q2 launches the Flyer without permission, telling Icheb he knows how to open a spatial flexure to another system. Icheb strongly objects, but Q2 wants to hide from his father. He breaks free of Voyager and jumps through the rift to the Clevari system. The Flyer is met by a Chokuzan ship, whose commander accuses them of trespassing and demands their surrender. Q2 fires on the ship and tries to escape into another flexure, but the Chokuzans fire back, and a bolt of energy knocks Icheb out. Q2 returns to Voyager to bring Icheb to Sickbay, but the Doctor can't treat him unless he knows more about the weapon that struck him. Q shows up again, and Q2 pleads with him to save his friend's life, but Q refuses, telling the boy he has to face the consequences of his actions.
To save Icheb, Q2 agrees to return with Janeway to the ship that he attacked. Q2 apologizes to the ship's commander and asks for his help, but when the commander threatens to hold Janeway accountable for the boy's actions, Q2 objects and insists on accepting punishment, even if it's torture or execution. The Chokuzan commander laughs and reveals himself to be Q in disguise. All along the Chokuzan incident was a test, which Q2 passed with flying colors. After Q assures Janeway that Icheb will make a miraculous recovery, the two Q's and Janeway appear before a tribunal of Q-Judges. Although Q argues that his son was willing to sacrifice himself to save another, the judges rule that Q2 has not made sufficient progress and must suffer the fate of staying human. Q is outraged, and disappears with the judges. Q2 dejectedly returns to Voyager, where he asks if he can stay and continue his training, feeling he still has a lot to prove to Janeway. But Q comes back and explains that he told the judges he wouldn't stay in the Continuum without his son, and claims they "begged" him not to leave. In fact, Q2's powers have already been restored. Q2 fills Janeway's ready room with flowers to thank her, and leaves on Q's promise to be a better father. Q reveals to Janeway that he had to agree to one minor condition—eternal custody of the child. In thanks, Q gives Janeway data on a shortcut home, but he won't take Voyager all the way because it would be a bad example for his son.