Continuity mistake: As Data, Geordi, and Will pause to consider a route around Engineering section, the smoky mist is chest high. After the angle change it's thinner and shin height.
Continuity mistake: On the freighter, when Geordi looks at Data, the image on the Bridge view screen has Data's hair in an unkempt, or unusual style. When the scene shifts back to the freighter, Data's hair is straight back.
Visible crew/equipment: When Worf addresses Klingon captain K'Nera, a microphone dips into the shot over Worf's head. (00:33:10)
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