Other mistake: In the transporter room, Data asks the computer what file log is shown. The computer reply is "Transport Log 759", Data confirms "That is the correct log", but the screen reads "Transport Log 721".
Plot hole: About two thirds of the way through this episode, Data deduces how Moriarty was able to 'leave' the holodeck. The big reveal is that he never did leave the holodeck, he merely reprogrammed it to simulate the rest of the ship without Picard, Barclay or Data's knowledge. While it is believable that this would fool humans like Picard and Barclay, it is ludicrous to suggest that Data would be taken in by it, even for a second. Data is an android whose perceptions of sight, sound and the world around him are far more sophisticated than humans. To list what we know of Data's perceptive abilities from previous episodes would take up the entire page, but suffice it to say he should have immediately recognised the 'Enterprise' as force fields and holograms rather than the genuine article. Note: this goes beyond a 'character mistake' or anything like that. Data's enthusiasm for Sherlock Holmes style deduction should have led him to eliminate the impossible before considering what was probable. Data accepts the impossibility of a holodeck character existing is the real world before discounting the possibility that it was still a holodeck simulation. This contradicts an awful lot of what is known about Data's abilities and powers of deduction. Although it serves for a good mystery the fact is it takes Data far longer than it should to deduce, or even guess at, the truth of their situation.
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