When Q first meets Amanda in her quarters, Picard witnesses this in his normal red uniform. Afterwards, Picard discusses this encounter with Q in his ready room. It's apparent from their conversation that this scene takes place a short time later, but Picard now wears his jacket and gray undershirt.
When Amanda is testing tricorders, a number of them are clearly engineering rather than medical tricorders (seen with the array of sensors on the front rather than the detachable silver medical scanning cylinder)
In Hide And Q (1987), the Q have the ability to give humans the powers and immortality of the Q as well as take them away. Q doesn't even suggest taking Amanda's Q powers away. However, Amanda isn't human; she is simply a Q in human form, as both of her parents were. Q does, however, mention (to his shadow-like colleague) that it may not be necessary for them to destroy Amanda.
Amanda and Q are playing Q's version of hide and seek. When they briefly appear in engineering, Data and Geordi see them, but don't bother reporting to the captain. People, and Qs, should not be free to go everywhere on the ship. However, even although Data and Geordi may well have reported it to the bridge afterwards, there is nothing anyone could do about it. The Enterprise crew know full well that Q can do whatever he wants to.
Obvious stunt double for Q when Amanda hurls him across the room.
The grass at the gazebo is laid out rolls of sod, instead of naturally grown, indicating that it's a set piece instead of a real location.
Q says that Amanda's parents were killed because they broke the agreement not to use their powers while living as humans, Thus the question arises as to why they did not use those powers to resist the effect of the tornado which killed them.
When Dr. Crusher has Amanda test the tricorders on herself, she holds the first one backwards.
Riker comes into the lab where Q and Amanda are and says he's looking for Dr. Crusher. He could have just used his comm badge.
Dr. Crusher tells Amanda that Nurse Ogawa would assist her take the tricorders down to the shuttlebay, yet we see her lugging two cases of tricorders on her own.
After the warp core breach, when Dr. Crusher addresses the senior staff, she enters the room and says (referring to Amanda) "He's a little shaken up, but she's gonna be fine."
Using his powers, Q takes himself and Captain Picard from the conference room to the captain's ready room. The rest of the people in the conference room don't know where they've gone. When Q and the captain are done talking and Q leaves, the captain doesn't bother telling those in the conference room that he is all right. He simply goes to the bridge and starts speaking to the crew that are there.