Amanda Rogers was a female Q born in 2351 in Topeka, Kansas, on Earth's North American continent. Unlike other Q, she was raised as a Human into adulthood when she joined the Q Continuum and consequently retained a moral code.
Her biological father and mother were two Q who assumed Human form to live on Earth. They were executed by the Q Continuum through a freak tornado accident in 2356 when Amanda was five. She was raised by adoptive parents, two Starfleet marine biologists. As such, Amanda did not know about her true Q nature and lived a Human life. However, she occasionally experienced objects come in and out of existence based on her desires. She never had it explained and simply lived with it, but tried not to let it affect her normal life. Increasingly, however, she gained the attention of the other Q, who became concerned about whether she was fully Q and potentially a danger if not brought into the Continuum.
Her academic accomplishments resulted in degrees in neurobiology, plasma dynamics, and eco-regeneration (with honors). She was accepted to Starfleet Academy in 2369 and selected out of hundreds of applicants for an internship on the USS Enterprise-D.
In 2369, she traveled to Starbase 112 where she transferred to the Enterprise-D to begin her internship. Doctor Beverly Crusher acted as her mentor, and arranged for her to be able to work in all the major departments. She was delighted to experience everything, but soon her unexplained Q nature increasingly exhibited itself when Q covertly tested her on her abilities. A litter of puppies appeared in Rogers' quarters in private, then she secretly saved Commander Riker from a falling container. Her secret was finally out when she saved the ship from a freak warp core breach.
After Q explained his intentions to test and train Rogers to the Enterprise crew, Amanda was informed of her true nature and was introduced to Q by Jean-Luc Picard and Beverly Crusher. She was shocked at the scope of what she was told, and, initially, decided to stay on the ship and learn not only her Enterprise duties but also her Q abilities. However, when she tried to mix them, by performing an experiment with those abilities without the due diligence it required, she realized they may not be able to coexist. Furthermore, she acted on a crush she had on Riker by taking him to a romantic fantasy only to realize it wasn't real after he pointed it out.
While this was going on, Picard decided to investigate Roger's parents' death and found out the truth. He confronted Q about it and told Rogers. Q then gave Rogers the same choice her parents were given. She initially decided to stay and refrain from using her powers, but then the Enterprise-D received a distress call from Commander Riker, who was organizing the Enterprise-D's relief mission on Tagra IV. A reactor was overloading, threatening thousands of people. When La Forge told them that he could not fix the problem and meltdown was imminent, Rogers intervened and the problem seemed to resolve itself. Even the planet's significant atmospheric contamination was restored to its natural state.
Acknowledging her true nature, Amanda told Q that she would go with him, but that first, she would need return to Earth to visit her adoptive parents and explain these events. She told Q, sternly, that he would have to be patient with her as she struggled to do this. Before leaving Rogers asked Dr. Crusher if she might come back one day to see her and then left the Enterprise-D with Q. (TNG: "True Q")
Background information Edit
Amanda, along with Junior, are the only children of Q known to exist.
In a 2011 interview, d'Abo commented, "Amanda had a vulnerability I related to because she was an orphan and in the episode was clearly at a stage in her life where she had some pretty important choices to make about her future. She had a strength and a wisdom about her. She was searching for the truth and searching for her own sense of truth. Though she was impressionable at times, it was inevitable to me through playing her that through her own self discovery she'd know which choices to make and what path to take." 
During the 0 Crisis, during which the USS Enterprise-E attempted to penetrate the galactic barrier, the female Q told Doctor Crusher that Amanda Rogers didn't "count" as a true child of the Q as she was conceived in a purely Human fashion, whereas "Junior" was conceived in the Q Continuum itself following "The Q and the Grey".
Amanda Rogers appears again in The Eternal Tide – referred to as "Q" by other Q – where she is shown to have formed a sibling-like relationship with "Junior". She is subsequently erased from history when trying to determine the source of an anomaly that is preventing her and Junior from traveling into the future by accessing an alternate timeline where USS Voyager was able to deal with the anomaly on its original trip home (before the events of "Endgame" changed history) only to be erased from existence as the resolution of the threat in that timeline also erased the Q from existence. Only Junior is shown to retain any memory of Amanda's existence.
An alternate fate for Amanda is presented in the miniseries Star Trek: The Q Conflict; after Amanda assists Wesley Crusher in giving the four captains the tools needed to end Q's conflict with other omnipotent races, part of the final truce sees Amanda and Wesley being stripped of their powers, Q assuring Picard at the end of the series that Amanda has been sent to begin a career in biomedical sciences.