Yes, it is possible to obtain the J-integral from an elastic-plastic analysis of a component containing a crack using Abaqus. An analysis will provide this output if you request a Contour integral via the Interaction module in CAE (Special > Crack > Create).
You may know that the J-integral is generally applicable to elastic materials, but only partially correct for elastic-plastic deformations. Problems can arise if the load applied is not monotonic. The most significant non monotonic loading occurs when a load is removed.