We present a new numerical algorithm that aligns a quadrilateral grid with internal alignment curves (IACs). These IACs can be used to delineate internal interfaces, discontinuities in material properties, internal boundaries, or major features of a flow field. The adaptive IAC grid generation algorithm readjusts a predefined reference grid to create a nearby grid where the mesh cell edges are aligned with the IACs. On an aligned grid, numerical discretizations of partial differential equations can be formulated to satisfy the interfacial relations, such as matching fluxes across the discontinuity, to reduce the numerical errors introduced by the discontinuity. We present examples to demonstrate the effectiveness of the IAC grid generation algorithm for multiple embedded interfaces in a quadrilateral grid.