I don't know what "60-100%" accurate means. Perhaps it would help to give a few examples of results that you regard "inaccurate" and the exact solution you were expecting.
I can give some generic information, however. QUAD2D has tolerances. You can use them to affect how much accuracy the result will have. When the (unknown) exact answer is q0, it's designed to return the approximation q when (approximately) abs(q-q0) <= max(abstol,reltol*abs(q)). If QUAD2D cannot give you this accuracy with confidence, it should warn you. What should not be happening (except in certain edge cases) is that it blithely returns a result that doesn't approximately meet the error tolerances without warning. Do you get any warnings?
There are usually things you can do to mitigate difficulties, for example, when there are discontinuities, placing them on boundaries by splitting an integral into multiple pieces. However, without more information about what your problem is, it is impossible to say whether QUAD2D can give better results or whether any other generic method would. Some problems do require special treatment.
