Where Silence Has Lease - S2-E2
Revealing mistake: When the black deck officer is supposed to be dead you can see the actor's nostrils flaring as he is breathing.
Revealing mistake: As Data cautions La Forge not to say anything, his actual flesh tone is visible on his palm.
Revealing mistake: When looking at Kolrami, the line between his make up and actual skin tone can be seen between his scalp, hairline, and the back of his neck.
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