On episode 3, Captain Lorca says that the Beta quadrant is 90 light-years away. 90 light-years is NOTHING in a galactic scale. The diameter of our galaxy is estimated at 100 000 light-years.
Of course it is possible that we are at the edge of the Alpha quadrant, but the fact remains that 90 ly is a very short distance.
You guessed right - we are at the edge of the Alpha/Beta border:
https://i.stack.imgur.com/C9YXQ.jpg
Since this takes place in the Prime Universe, the Beta Quadrant border being 90 light years away is extremely plausible, as the border runs right through the Federation. Maps of TNG, DS9, VOY, and even ENT also agree with this. Star Trek Online agrees with this.
Klingon Space is within the Beta Quadrant, as is the Romulan Star Empire.