Projections - S2-E3
Other mistake: At the beginning of the episode, as B'Elanna is working her way out through the doors, the Doctor cowers and tries to hide behind things. There is no reason for him to be afraid. He can't be hurt under normal conditions - this proves to be the case later when he feels pain and says he is not programmed to feel pain, and we learned earlier in "Phage" that he can change whether or not things pass through him.
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 ★