Plot hole: It's stated that only objects or people infused with the chronoton serum can pass through the time barriers, which is why the equipment belts and hyposprays had to be specially modified (these wouldn't be needed if the chronotons were producing a protective field wider than the infused object). But some un-infused artificial objects, such as clothing and com-badges, can pass through them without vanishing. This is most obviously seen in the scene when Chakotay pulls the captain through the barrier, since he doesn't even wait long enough for the serum to spread throughout her entire body, yet alone to perspire into her uniform.
Plot hole: Tuvok says that he was born on Stardate 38774. But he was born in 2264, when Stardates were only counted in the thousands, not the tens of thousands. If he were really born on SD 38774, that would mean he was born only a few years before the Enterprise-D was launched.
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 ★