No, the Borg had already been probing the Federation and the Romulan Star Empire before that encounter.
Along the Neutral Zone various Federation and Romulan colonies disappeared, whole cities just scooped right up.
Q’s interference may have caused the Borg to attempt assimilating Earth much earlier than intended, but the Borg were already on their way and they had no idea.
Due to Q’s interference, however, the Federation was made aware of the Borg well before a Cube came on their doorstep.