(covers information from several alternate timelines)
In 2259 in the alternate reality, Kirk confiscated a K'normian trading ship during the Mudd Incident a month before using it to infiltrate Qo'noS, while themselves posing as K'normian arms dealers. (Star Trek Into Darkness)
The K'normians were designed by Fred Phillips and Robert Fletcher for The Motion Picture. With the approval of Gene Roddenberry, the K'normians, as well as numerous other new species in the film, were named by Fletcher and provided a backstory by him. The following is a brief description, in Fletcher's own words:
"K'NORMIANS – From planet K'norm. Similar to Earthlings but have additional brain structure. Additional part of brain performs functions we aren't capable of, mainly for long-distance communication. They have an eighth sense, can deal with time and space dimension." (The Making of Star Trek: The Motion Picture, p. 130)
In an interview with Robert Fletcher – published in the February 1980 edition of Fantastic Films, as part one of a two-part interview with him, called "The Star Trek Costumes" – the name of this species was misspelled "K'norian" and they were described thus;
"Similar to Earthlings in all aspects save one: encased in a type of structural shell formation on top of the forehead is an additional brain structure enabling them to communicate over very long distances. Included in this brain structure is, in essence, an eighth sense, giving them the ability to deal with time and space dimensions – an ability we Earthlings sadly lack." 
In an article appearing in the January 2002 edition of Star Trek: The Magazine Volume 2, Issue 9, entitled "Who is that Alien?", the following information about the K'normians, as taken from Fletcher's original background notes, added:
"The K'normians were a mostly normal humanoid race, though they did have large foreheads not unlike the Klingons, which Fred Phillips also redesigned for The Motion Picture. The K'normians' large brains provided an eighth sense that gave them a superior appreciation of time and space."
According to Star Trek: Star Charts ("United Federation of Planets I", "United Federation of Planets III") and Stellar Cartography: The Starfleet Reference Library ("Federation Historical Highlights, 2161-2385"), there was a planet named K'normia in the star system K'normia (Psi Tauri), which might possibly be the system of origin for this species, in the Alpha Quadrant. This system's primary was a F-class star. In 2378, K'normia was listed as a Federation member.