TOS used the terms "quadrant" and "sector" interchangeably, but one could argue--as proposed by some--that space was so much bigger in Kirk's day that sectors were divided into quadrants. By the time of the Khitomer Conference, however, this practice of having "sector quadrants" had likely ended.
You are so two-dimensional
Who is to say the Klingons had to traverse federation space to get there?
That's a valid point. Lose the idea that each galactic government exists in a perfect sphere, it's easy for them to have territories that wrap around one another.
There is enough onscreen material to piece together an idea that the Federation and some of the other local galactic governments in the Trek Universe all occupy a region that spans the line where the Alpha and Beta Quadrants meet, with parts of their borders fairly close to one another. Having so many interstellar powers in close proximity to one another along the Alpha-Beta Quadrant border would explain some of the conflicts the Federation gets into.
We could still have it be a case that the Klingons are mostly
in the Beta Quadrant but have some territory or colonies in the Alpha Quadrant as well. There could also be regions of non-aligned space that anyone can cross through (a Neutral Zone could just be an agreement prohibiting Federation ships from crossing into Klingon territory or Romulan ships from entering Federation territory, but doesn't prevent any of them from crossing the Alpha-Beta Quadrant border through unincorporated territory).