Continuity mistake: When Geordi falls down the hole in the beginning of the episode his visor is seen to his right. When he wakes up he finds it to his left.
Continuity mistake: After Wesley makes his neutrino pulse suggestion, Picard tells him to go do it. Wesley walks off camera. Then there's a commercial break, and Wesley is back on the bridge in the long shots. A few minutes later we hear the turbo-lift and Wesley walks back onto the bridge.
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