Plot summary
The Federation starship Enterprise arrives at the planet Bre'el IV, whose moon is descending from its orbit and threatening to crash into the highly populated world below. As the crew scrambles to find a solution, Q suddenly appears on the bridge naked. He explains that he has come to the Enterprise seeking asylum, as he has been stripped of his powers and banished from his home dimension, the Q Continuum, as punishment for his chaotic antics. Captain Picard is skeptical of Q's story but reluctantly offers him shelter, albeit in the brig.
Picard urges Q to use his powers to return the moon to its proper orbit. Q insists he is powerless but offers his vast intellect and experience to assist the crew. Picard agrees and assigns Lt. Commander Data to monitor him. In engineering, Q flippantly suggests altering the gravitational constant of the universe, inspiring Chief Engineer La Forge to attempt a similar effect by wrapping the moon in a low-level warp field, allowing the ship to drag it back into place.
Meanwhile, Q struggles to adapt to his newfound humanity, experiencing fatigue, hunger, and other human conditions. Data tells Q that, as an android, he aspires to have the human experiences Q now resents. Shortly after, Q is attacked by the Calamarain, a gaseous alien species he once tormented. Picard deduces that Q sought refuge on the Enterprise to protect himself from the countless creatures he has mistreated. When the shields are lowered for a test of La Forge's procedure, the Calamarain attack again, and Data is nearly electrocuted defending Q from their assault. Realizing that his presence is causing more harm than good, Q steals a shuttlecraft and leaves the ship, intending to sacrifice himself to draw the Calamarain away.
As the Calamarain close in on the shuttle, a second Q being appears and informs Q that due to his selfless act, the Continuum is willing to restore his powers. Q accepts and subdues the Calamarain, but releases them after being reminded to reflect on the lessons he has learned. He then returns to the Enterprise to bestow a parting gift on Data: a few moments of laughter.
On learning that the Bre'el moon has been safely returned to orbit, Picard surmises that Q was responsible and muses that the omnipotent being may have a residue of humanity after all. However, Q's voice reminds him, "Don't bet on it."
Credits
Adapted from Wikipedia contributors, Deja Q. Paragraphs and formatting adjusted; illustrations, reference markers, block quotations and non-story material omitted.