In order to further improve the research and development of image edge detection algorithm on quantum computer, a quantum image binary morphological edge detection algorithm is proposed combining with quantum computing operation on IBM Q platform. Based on the NEQR quantum binary image representation and quantum binary etching and expansion operations, the corresponding binary morphologic smoothing filtering quantum circuit and edge extraction quantum circuit are designed and implemented. And then through Qiskit extension, the binary morphological edge detection of quantum images with sizes of 8×8 and 128×128 is realized on the virtual local quantum simulator of IBM Q platform. Meanwhile, the comparative simulation experiment of quantum images with a size of 8×8 is also realized on the real cloud quantum simulator of IBM Q platform. The experimental results prove the feasibility and effectiveness of the binary morphological edge detection algorithm for quantum image.