Within the film itself, it's made pretty clear that Picard can indeed hear the Borg. We see him tilting his head to hear the voices in his head, then he immediately orders an attack on a non-obvious weakness.
In the film's Paramount-licensed novelisation, the incident is made even more explicit:
In the millisecond before the android replied, Picard realized that he
inexplicably knew —
knew—the information he had requested, for his question had already
been answered by a whisper in his own head—a voice that was one, yet
many; a voice that evoked the ghost of a half-remembered feminine
Without thinking, he rose, entranced, and moved toward the viewscreen
where the image of the massive and unlovely vessel hung. They were
there; he could sense them, hear them speak. For an instant, he felt
as though he had only to reach toward the screen, and he would touch
The whisper of the one and the multitude grew briefly louder.
"critical damage to shields at power sector one-one one. All drones
coordinate repair immediately...."
The mental whisper died abruptly, as if the speakers had realized he
was listening. But it was too late; he had already experienced a
revelation beyond the mere words he had detected.
They were wounded. They were vulnerable, and he knew beyond all reason
the precise spot.