As early as Stardate 9521.6 - calendar year 2293 - Starfleet has begun a database on the Borg. Being highly classified, the extent of the information at the time and its sources are unknown. Perhaps it is a result of contact with the El-Aurians, who were rescued by the Enterprise-B:

from Voyager: Scorpion Part I.

On the basis of such rumors and hearsay, exobiologists Magnus and Erin Hansen petition and gain approval from the Federation Council on Exobiology to seek out the so-called Borg. The Hansens travel deep into the Beta Quadrant on their science vessel, the U.S.S. Raven, until they disappear (11 months after launch). Their daughter will later be called Seven-Of-Nine and will serve on the USS Voyager.

But the Borgs are now in contact with the federation and at least the Federation starship U.S.S. Tombaugh, commanded by Captain Blackwood, is assimilated 2362. (see Voyager: Infinite Regress).

That is why you can forget Q, who moved the Enterprise-D 7000 light-years from Federation space away and that stuff from the year 2365...