We investigate theoretically the ability of graphene nanopore membranes to detect methylated sites along a DNA molecule by electronic sheet current along the two-dimensional (2D) materials. Special emphasis is placed on the detection sensitivity changes due to pore size) shape, position; and the presence of defects around the nanopore in a membrane with constricted geometry. Enhanced sensitivity for detecting methylated CpG sites, labeled by methyl-CpG binding domain (MBD) proteins along, a DNA molecule, is obtained for electronic transport through graphene midgap states caused by the constriction. A large square deviation from the,graphene conductance with respect to the open. nanopore is Observed during the translocation of MBD proteins. This approach exhibits superior resolution in the detection of multiple methylated sites along the DNA compared to conventional ionic current blockade techniques.