To get an accurate aggregation we must start at `{20120101}`:

     data = {{{20120101}, Missing[]}, {{20120802}, 193.9}, {{20120912}, 
       493.9}, {{20130412}, 33.9}, {{20130502}, 193.9}, {{20130802}, 
       193.9}, {{20130822}, 193.9}, {{20131002}, 193.9}, {{20131022}, 193.9}};

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

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

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

     
    DateListPlot[
     TimeSeriesAggregate[Transpose[{dates, data[[All, 2]]}], "Quarter", Length],
     DateTicksFormat -> {"Year", "/", "QuarterNameShort"},
     GridLines -> Automatic,
     PlotRange -> {{{2012, 8, 1}, {2013, 12, 31}}, Automatic},
     Mesh -> Full]

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


And here are the monthly figures

    DateListPlot[
     TimeSeriesAggregate[Transpose[{dates, data[[All, 2]]}], "Month", Length],
     DateTicksFormat -> {"Year", "/", "Month"},
     GridLines -> Automatic,
     PlotRange -> {{{2012, 8, 1}, {2013, 10, 31}}, Automatic},
     Mesh -> Full]

[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/d7geR.jpg
  [2]: https://i.sstatic.net/ewzTQ.jpg