(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.
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. (TNG: "Sins of the Father"; Star Trek VI: The Undiscovered Country)