Other mistake: When Data is interpreting between Captain Picard and Riva, he uses signs for words Captain Picard has not yet spoken, and voices concepts before Riva has signed them. He also doesn't interpret everything the Captain and counselor Troi say.
Audio problem: After the offer of refreshments has been declined, Harmony says "I have been expecting you", but her mouth doesn't move.
Visible crew/equipment: After Picard, Troi, and Worf have beamed down to the planet to meet Riva, when Harmony asks if they need food or refreshments, just as Picards says, "Thank you, no," a crewmember's elbow can be seen at the left side of the screen (this one is not visible on the Netflix version). Then when Troi says, "Your method of communication is elegant and quite beautiful," a crewmember (likely the same one as earlier) can be seen at the left side of the screen. (00:07:20 - 00:10:40)
Continuity mistake: Near the end of the episode, as they are preparing to beam down to the planet, Riker tells Warf to set his phaser on stun. He then holsters his own phaser and the transporter is activated. However, when they materialize on the planet, Riker is holding his phaser in his left hand.
Audio problem: We hear Reva's female member of his choir say "I have been expecting you", but her mouth doesn't move.
Answer: He brought the Borg to the Alpha Quadrant and showed them that it was full of worlds waiting to be assimilated. Guinan's homeworld was their first stop, and they assimilated everyone and took over the planet, leaving The Survivors of her race without a home. Q is ultimately responsible for that.
Captain Defenestrator
By the time Q takes the Enterprise to meet the Borg, Guinan already knew who they were and they had already destroyed her world. Therefore the above answer can not be right. I believe Guinan is much more than she appears, and her people have had encounters with the Q in the past. It is these interactions, that obviously were not pleasant, that fuels her distrust.
oldbaldyone
That's what the above answer is saying. Q brought the Borg to the Alpha Quadrant (not Earth) and the Borg destroyed Guinan's home world in the late 2200's, which is why she hates Q. Although she met Q in 2160 and they both saw each other as enemies right away.
Bishop73