The Q Civil War was a conflict between two factions of the Q Continuum that took place in 2373. The deeper nuances of the conflict were beyond the understanding of "mere" Humans, who perceived the conflict in the format of the American Civil War. The war was apparently fought between a faction interested in the status quo of the Continuum, and a faction interested in disrupting the static order of the Qs' existence. The latter faction was led by Q.
The war first broke out following the suicide of Quinn in 2372 (cf. VOY: "Death Wish") – the first death of a Q in many years, and an event which severely disrupted the sense of order of the Continuum. Although Q had previously been a supporter of the status quo faction, Quinn's death, and his ideas of freedom and individualism, convinced him to switch sides and become a proponent of less order and control over the actions of individual Qs. Q himself started the conflict by "sounding the trumpet and carrying the banner." The other Q's who believed in Quinn's ideas followed him. When the status quo Qs tried to destroy them, they fought back, and the war spread quickly.
One of the side effects of the civil war was a series of supernovae that erupted across the Delta Quadrant within the space of a mere few days; the occurrence of multiple supernovae only a handful of light years from each other in such a short time span was unheard of, and believed to be caused by the use of some type of Q weapon as part of the civil war.
Seeking to bring an end to the conflict, Q proposed conceiving a child with Kathryn Janeway, whom he had met the previous year. Q hoped that a Q/Human hybrid child would unite the best qualities of both species and restore the balance of order in the Continuum on a level that both sides would accept. However, Janeway refused to accept Q's plan, arguing that, aside from the fact she did not want to mate with him, his plan, though noble, offered only a "quick fix" without addressing the underlying problems that the conflict was about.
Janeway then attempted to negotiate a truce to end the civil war by contacting a leader of the opposition faction, Colonel Q. However, the status quo faction refused to negotiate, and instead planned to end the conflict by executing Q and Janeway. Before the executions could be carried out, however, the pair were rescued by personnel from Voyager, who were assisted by a female Q, an old flame of Q's.
Q then made his proposal of conceiving a child to his old girlfriend, and she accepted. The two touched fingers, and in a small glow of energy created the first new Q in over ten millennia. This newborn child (nicknamed Junior) was hoped to be a herald of a whole new era of peace for the Continuum, and brought about a quick end to the civil war. In gratitude for her assistance, Q named Janeway the "godmother" of his new son. (VOY: "The Q and the Grey")