The Royale - S2-E12
Trivia: In this episode, Picard is studying Fermat's Great Theorem, and says it has remained unsolved for 800 years. Five years after the episode was made the theorem was proven, by Andrew Wiles and Richard Taylor from Princeton University (their proof is not the same as Fermat's though, as they used modern methods Fermat did not know of). In the Star Trek universe, this was referred to in an episode of Deep Space Nine, and is considered as a subtle correction for Picard's statements.
Trivia: Closed captioning, in its infancy in the 80s, often dropped words and letters by accident. In the original broadcast of this episode, the captioning of Riva's line, "We could dine together," lost an N, resulting in a rather bizarre exchange. Riva: We could die together. Troi: I'd like that.
Trivia: As Picard argues for Data, the wall behind Maddox can easily be seen as the forward wall and viewer of the Enterprise Battle Bridge, re-purposed for this episode.
The Royale - S2-E12
Trivia: When Captain Picard is reading Hotel Royale in his ready room, he comments that the book's first line, 'it was a dark and stormy night', is "not a promising beginning". This line is actually the first line of Edward Bulwer-Lytton's Paul Clifford; the line has become so clichéd that it frequently appears in satirical works to denote humorously incompetent or overly melodramatic writing.
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