This study aims to estimate the quantum Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy in curvilinear coordinates. We used the results to calculate the quantum binary and triplet distribution functions in curvilinear coordinates. The analytical form of the quantum distribution functions was obtained for dusty plasma in Saturn's rings model. We use particles-in-cell (PIC) simulations to find a visualization of dusty three-component plasma phase space in curvilinear coordinates. Our results were compared with others.