Not necessarily. It was in The Next Generation episode 1.4, "The Last Outpost" that the Enterprise had an official first look at the Ferengi, a race of pirates about whom very little else was known at the time.
This probably means that there was never a confirmed meeting with the Ferengi on record in all those centuries before, or no crew member had ever knowingly met one. When Captain Archer had a run-in with them, we as the audience know they are Ferengi, but they never identify themselves as such to the crew. So Archer could never made an exact record in the ship's logs about these intruders, so there never was an official First Contact. This can be compared to the episode "Carbon Creek", where T'Pol reveals that the humans had already met the Vulcans without knowing it many years before official First Contact.
Furthermore, Archer warned the Ferengi to stay away from Starfleet vessels, so this may have further limited human contact with them for the years to come.
So in short, Archer's meeting with the Ferengi does not 'count' as a real First Contact.