The Beta Stromgren system was an uninhabited star system in Sector 25434. The system was in space claimed by the Romulan Star Empire.
A Federation long-range unmanned probe, Vega Nine probe, had charted this system, which lay twenty-three parsecs beyond the limits of manned exploration at the time. This system was destroyed in 2366 when its primary, Beta Stromgren, became a supernova. (TNG: "Tin Man")
"STRUM-gren" was the pronunciation for the system's name from the script pronunciation guide. [1]
Based on the system's seemingly close proximity to Starbase 152, it would seem that this system is in, or is relatively near the Beta Quadrant.
According to Star Trek: Star Charts ("United Federation of Planets II") and StarTrek.com, the Beta Stromgren system was located in the Beta Quadrant. [2]