They did (warning, dubious canon ahead)
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In the Trek novels "[Mere Mortals][1]" and "[A Singular Destiny][2]" the Borg send [a fleet][3] of over 7000 cubes to attempt to assimilate the Alpha Quadrant. At the cost of thousands of vessels they eventually establish a toe-hold in the quadrant and, before their eventual destruction (courtesy of the [Caeliar][4]) they wreak havok on Klingon empire and kill over 63 billion sentients. 


  [1]: http://en.memory-alpha.org/wiki/Mere_Mortals
  [2]: http://en.memory-alpha.org/wiki/A_Singular_Destiny
  [3]: http://memory-beta.wikia.com/wiki/Borg_history
  [4]: http://memory-beta.wikia.com/wiki/Caeliar