Star Trek: 10 Secrets Of The Q Continuum

The Q Civil War was caused by the suicide of one of their people, so Q hoped that he could end the conflict by creating a new life, something which he claimed had not been done in the Continuum for over ten millennia (presumably, since Amanda Rogers was born on Earth and raised as a Human). He originally hoped to mate with Captain Janeway, a Human, in order to combine the best parts of both their species', but when Janeway refused, he instead convinced his old partner (another Q) to conceive a child with him.

Q and his partner discussed how to conceive a child, and decided to touch their fingers together, producing a little spark of light, which apparently got the job done.

The child, who became known as Q Junior, was meant to bring peace to the Continuum and usher in a new era, but he became more and more unruly as he approached adolescence, starting war, detonating Omega molecules, and knocking planets out of their orbits. Q came to Voyager one last time in the episode Q2, in the hopes that Janeway would be able to straighten out the boy (who now appeared as a teenager, because of how time worked in the Continuum). Q Junior caused a lot of chaos on the ship, and eventually endangered Icheb's life. This devastated Junior, and he chose to sacrifice his own life to save his new friend, only for Q to reveal that he caused Icheb's injury in order to teach his son a lesson.

Because of his offer to sacrifice himself, the Continuum chose not to turn the boy into an amoeba (as they originally planned), but instead into the next worst thing they could think of: a Human. However, Junior's punishment was short-lived, and the Continuum soon came to the decision to allow him to keep his powers, under the strict and constant supervision of his father. Apparently, this sentencing must have been finished by Star Trek: Picard season two, as we saw no mention of him after this episode.