Miss Q was a member of the Q Continuum and 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 Qs' 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, when her son, Q, 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?!"
"Never mind that. What ARE you doing with that dog?"
- - Q (male) and Q (female), looking at the dog Janeway is holding
"I'm not talking about the puppy."
- - Q (female)
In the Q Continuum novel trilogy, Q visits the USS Enterprise-E along with the rest of her family and attempts to halt a Federation experiment to penetrate the galactic barrier using an artificial wormhole. During their time on the ship, she reflects on her own fears about motherhood, but is slightly consoled by Doctor Beverly Crusher. Her son is briefly captured by an insane Betazoid scientist who has been enhanced by the barrier's psychic power and subsequently "possessed" by Q's old enemy/mentor 0, the scientist intending to test Junior's capabilities in various ways, but Q manages to get her son released.