I note that all answers so far try to solve the problem of assigning a potential Null value by manipulating the return value. I feel it would be more appropriate to make the whole assignment conditional. Like this:
If[condition, aa = value]
There's also a small bug in your program (count isn't initialized), and, of course, it doesn't sort at the moment. I assume that you aware of that and that the Return value is used for testing.
The code would then be:
QuickSort[x_List] :=
Module[{pivot, aa = 0, bb = 0, count = 0},
If[Length@x <= 1, Return[x]];
pivot = First[x];
If[Length[Cases[x, j_ /; j < pivot]] > 1,
aa = Length[Cases[x, j_ /; j < pivot]] - 1
];
If[Length[Cases[x, j_ /; j < pivot]] > 1,
bb = Length[Cases[x, j_ /; j > pivot]] - 1
];
count = count + aa + bb;
Flatten@{QuickSort[Cases[x, j_ /; j < pivot]],
Cases[x, j_ /; j == pivot], QuickSort[Cases[x, j_ /; j > pivot]]};
Return[count]]
Remove ;Return[count] to let it sort again.
I'm not sure about the j < pivot test in the second If. It depends on your intention with bb, but I guess the test should be j > pivot.