(written from a Production point of view)
The name Klingon system has only been referenced once in a cut scene in TNG: "Sins of the Father" by Kurn. The capital of the Klingon Empire, the planet Qo'noS, was located in this system. ENT: "Two Days and Two Nights" establish that Qo'noS, was less than 90 light years from the Sol system.
The star of the system was seen in TNG: "Redemption" as the USS Enterprise-D and IKS Bortas approached Qo'noS. It has not been named in canon. According to the Starfleet Medical Reference Manual, the Klingon homeworld was the planet "Epsilon Sagittarii B". Epsilon Sagittarii is a real star located 143 light years away from Earth. According to the Star Trek: Star Charts (page 56), the star was an orange dwarf. Star Trek Maps and The Worlds of the Federation name the star as "Klingon".