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

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enter image description here Source

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

So, to answer your question: most of the episodes of TOS, TNG, DS9 and ENT (and now DIS too) take place more or less equally both on the Alpha and the Beta quadrants.