The Beta Quadrant was the common designation during the 24th century for one-quarter of the Milky Way Galaxy when viewed from a point of view perpendicular to the plane of the galaxy. It extended ninety degrees spinward of an imaginary line from the center of the galaxy through Sector 001 (which contains the Sol System.)
The Typhon Expanse, Deep Space 4 and Bolarus IX were all located in the Beta Quadrant, as were Myhr'an, Wthaure and Thrieshok-nar space. (Star Trek: Star Charts; Star Trek: The Cantabrian Expeditions)
The majority portions of the Klingon Empire and the Romulan Star Empire were located in the Beta Quadrant. The Federation also claimed territory in that Quadrant in the 23rd and 24th centuries, although it was more frequently identified during that time as an Alpha Quadrant power.
The Federation knew little of those portions of the Beta Quadrant not already claimed by known powers well into the 24th Century.
Between 2250 and 2258, the USS Antares explored the near reaches of the Beta Quadrant, approximate to Federation space. (Ship Recognition Manual, Volume 4: Starships of the Original Series Era)
In 2293 the starship USS Excelsior, commanded by Captain Sulu, completed a three year exploratory voyage in the Beta Quadrant which included cataloguing gaseous anomalies. Seventy years later Captain Lisa Cusak and the crew of the USS Olympia began a seven year exploration voyage in the Beta Quadrant.
In 2372, the crew of the USS Cantabrian assumed the rise in Myhr'an violence was due to a "sponsor" race trying to destabilize the Beta Quadrant. (Star Trek: The Cantabrian Expeditions: "Catalyst, Part One")