In 2356, the Haakonians conquered Talax following a decade-long war which concluded with the Haakonians' deployment of the metreon cascade on the Talaxian moon, Rinax. This weapon caused such devastation that it instantly ended the war, with the Talaxians yielding their homeworld and colonies to the Haakonians.
The only known Federation encounter with the Haakonians was made by the USS Voyager in 2371, when the Haakonian scientist Ma'Bor Jetrel came aboard Voyager seeking Neelix. Jetrel had been involved in the war with the Talaxians, having developed the metreon cascade which killed hundreds of thousands of Talaxians. Although ostensibly seeking to cure a side-effect of the cascade, Jetrel was actually trying to reverse its original effects by using Voyager's transporter systems to re-integrate victims that had been vaporized by the metreon cascade. Sadly his experiment failed and Jetrel died of metreon poisoning soon afterwards. (VOY: "Jetrel")
In 2376, Abaddon claimed that he had acquired a ship, later named Alice, from a Haakonian trader, who claimed that the ship was "haunted". Abaddon blamed the stories on isolation sickness and took the ship off his hands. (VOY: "Alice")
The Haakonian makeup consisted of heavy eyebrow appliances with ridges running up the sides of their temples to the hairline, a sharp appliance over the ridge of the nose, and a chin piece. (Star Trek: Aliens & Artifacts, p.169)
In the Star Trek Encyclopedia (4th ed., vol. 1, p. 327), the trader from "Alice" was identified as a member of the Harkonian species.