Corrected entry: Both the Borg and Chakotay reference that it would take months for Voyager to cross Borg space. The Borg use trans-warp regularly, which is the method Voyager eventually arrives home at the end of the series. The Borg could have scooped up Voyager and had them not only on the other side of Borg space, but all the way home in less than a day. Janeway really missed an opportunity in her negotiations.
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Correction: Janeway didn't know that the Borg had transwarp conduits that reached the Alpha Quadrant. Additionally, she would not want to encourage the Borg to send any ships towards Earth, since it would simply motivate them to invade. Janeway compromised - they would cross Borg space safely, and without leading to a Borg invasion of the Federation.
Yes she would have known the Borg had trans warp conduits to the Alpha Quadrant, as the Borg reached the Romulan Neutral Zone in TNG Season 1, the attack at Wolf 359 only a year after official first contact, and the rogue Borg were attacking Federation colonies in 2369, two years before Voyager was lost in the Badlands. Therefore, the Borg has a way to get to the Alpha Quadrant in very little time, which Janeway would know. Why the Borg didn't use the transwarp conduit that opened up right on earth's doorstep ("Endgame") is another question.
The Borg would not need transwarp conduits to reach the Alpha Quadrant, they could travel at regular warp speeds and although it may take them a longer time, they would still get there. It is never stated how long it took the Borg to reach the Neutral Zone or Wolf 359, however Q says that the Borg evolved over thousands of centuries.