A couple of maps for you, both taken from the *[Star Trek: Star Charts][1]* book:

>[![enter image description here][2]][2]
[Source][3]

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>[![enter image description here][4]][4]
[Source][5]

As you can see, the border between Alpha and Beta quadrants passes through the Solar System, effectively splitting the UFP territory in two almost equal parts. It is roughly a situation equivalent to the [Greenwich meridian][6]

**So, to answer your question: TOS, TNG, DS9 and ENT take place equally both on the Alpha *and* the Beta quadrants.**


  [1]: http://memory-alpha.wikia.com/wiki/Star_Trek:_Star_Charts
  [2]: https://i.stack.imgur.com/HEGgZ.jpg
  [3]: https://drofdemonology.deviantart.com/art/Star-Trek-Map-1-482984975
  [4]: https://i.stack.imgur.com/ODWaE.jpg
  [5]: https://drofdemonology.deviantart.com/art/Star-Trek-Map-2-509478876
  [6]: https://en.wikipedia.org/wiki/Prime_meridian_(Greenwich)