Q, as all members of the Q Continuum are called, was an old flame of Q's. The two had been involved in an "eternal" relationship for the past four billion years, although Q claimed he never said it was exclusive. During the Q Civil War in 2373, she chastised Q for offering to mate with Kathryn Janeway of the USS Voyager, a "mere mortal" Human.
Q was trapped on Voyager when the ship was battered by the shock waves of a pair of supernovae, which were side effects from the Q's use of weaponry in the civil war. Anxious to catch up with her lover, who had escaped with Janeway into the Continuum, she provided the Voyager crew with information that allowed them to enter the Continuum themselves. She also provided them with advanced weaponry that allowed them to rescue Q and Janeway from the custody of the status quo faction, which was planning to execute the pair.
At Janeway's suggestion, Q offered to conceive a child with his old flame as a means of ending the civil war, by bringing into existence the first newborn Q in over ten millennia. Q agreed, and the pair went off together to raise their new son. (VOY: "The Q and the Grey")
Unfortunately, Q Junior failed to live up to his parents' expectations; he became a notorious hell-raiser instead. Fed up with Q's lack of responsibility and their inability to control their child, she disowned her child and left Q Junior's discipline up to Q in 2378. (VOY: "Q2")
"Q! How did you find me?!"
"It is of no matter. What ARE you doing with that dog?"
(Q (male) looks at the dog Janeway is holding)
"I'm not talking about the puppy."
- -Q and female Q