Star Trek: Voyager (1995)

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 ★

Answer: Add to that the two transport ships at the start of Star Trek Generations were carrying El-Aurian refugees to Earth. It wasn't stated in the film what they were refugees of, but Guinan would state in TNG that the Borg wiped out her planet and most of people, so it's a safe bet that's what it was. And with 47 El-Aurians being rescued by the Enterprise-B, there were plenty of people to tell Starfleet about this cybernetic threat. At the time though, Starfleet did not have the ability or resources to investigate this further, and it was eventually forgotten when other things became important until the Enterprise-D encountered that cube at J-25.