Plot hole: To escape the 'event horizon', a rupture in it is widened to 120 meters (which allows 2 meters clearance on each side of the ship). Soon after, the rupture has shrunk to 110 meters wide, and Janeway decides to 'punch through' anyway. The ship does so with little difficulty, but the whole issue seems pointless since the rupture was far larger 'vertically' than 'horizontally' (relative to the view from the ship), so tipping the ship on its side would have allowed plenty of clearance to slip right through.
Factual error: The event horizon of a quantum singularity (aka black hole) is not a physical barrier you can punch a hole through but a theoretical one, which can't be escaped from without faster then light travel.
Chosen answer: Before Q sent the Enterprise to the beta quadrant to officially contact the Borg, there were already indications that the Borg was beginning to reach Federation territory. There were remarks towards the end of the first season of the Next Generation that several of the furthest Federation outposts were being attacked by some unknown enemy. They suspected the Romulans, but when contact with the Romulans was re-established, they learned that it was not them. The Hansens had simply figured things out much earlier than anyone else in the Federation. They learned about the Borg nine years earlier, but Starfleet mainly took notice when their outposts started getting wiped out. It is logical to assume that there were indications of Borg scouting parties and research efforts well before that.
Garlonuss ★