(written from a Production point of view)
The term "Klingon system" was only referenced once, by Kurn in "Sins of the Father", in a cut scene. The star, which was seen as the USS Enterprise-D and IKS Bortas approached Qo'noS in "Redemption", was never given a formal name.
TNG: "Sins of the Father", Star Trek VI: The Undiscovered Country, ENT: "Broken Bow", "Two Days and Two Nights" establish that the system, located within the Alpha Quadrant, was less than 90 light years from the Sol system. The capital of the Klingon Empire, the planet Qo'noS, was located in this system.
According to the Star Trek: Star Charts (page 56), the star was an orange dwarf. Star Trek Maps names the star as "Klingon". The star is called "K'lai Klinzhai" in FASA's The Klingons and "K'thar" Decipher's Worlds.