data =
     {{{20120802}, 193.9}, {{20120912}, 493.9}, {{20130412}, 
        33.9}, {{20130502}, 193.9}, {{20130802}, 193.9}, {{20130822}, 
        193.9}, {{20131002}, 193.9}, {{20131022}, 193.9}};

First we must transform the dates in a Mathematica-like form:

    dates = 
      ToExpression[{StringTake[#, 4], StringTake[#, {5, 6}], 
          StringTake[#, -2]}] & /@ (ToString /@ Flatten@data[[All, 1]]);

Then we use `TimeseriesAggregate` to count the frequency per quarter

      DateListPlot[
        TimeSeriesAggregate[Transpose[{dates, data[[All, 2]]}], "Quarter", Length],
          DateTicksFormat -> {"Year", "/", "QuarterNameShort"},
          Mesh -> Full]

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/CznkI.jpg