The traditional optical differentiators utilize interference, diffraction, or focusing properties to differentiate light waves, often requiring a complex optical setup for practical implementation. We propose a novel theoretical framework for an optical differentiator that achieves edge enhancement by precisely manipulating the orbital angular momentum (OAM) of pump light. By modulating the OAM of the pump light, higher-order differentiation of parametrically down-converted light waves is realized. Furthermore, leveraging the correlation properties of generated photon pairs by spontaneous parametric down-conversion (SPDC), we demonstrate quantum correlation imaging with edge enhancement. Based on quantum entanglement-based correlated imaging which enables non-local ghost imaging modality, we incorporate a high-order differentiator into the correlated imaging system to efficiently extract key features of images, thereby achieving non-local quantum image processing. The combination of these two techniques establishes a flexible and high-performance quantum optical differentiation system, providing a new technical pathway for optical computing and holding potential applications in future quantum image processing.