On several occasions in 2364, viewscreen readouts aboard the USS Enterprise-D showed the location of Barnard's Star, in the form of charts from the Enterprise-D library computer. (TNG: "The Naked Now", "The Last Outpost", "Conspiracy")
This chart naming stars within 20 light years of Sol was drawn by Rick Sternbach for the Star Trek Spaceflight Chronology in the late 1970s. Found on page 77, this chart showed Earth commercial and exploration routes after the use of warp drive began. Barnard's Star was a destination on an exploration route. The stars Alpha Centauri, Sol, Sirius, and Tau Ceti were identified on "The Explored Galaxy" wall chart as being located in the Alpha Quadrant.
Barnard's Star was located in the Alpha Quadrant. This was a binary star system. Barnard's Star A was a Class M star with a magnitude of +7, which was 1/10 as bright as Sol. Barnard's Star B was a Class M star. (Page 44)