By utilizing the tight-binding method, the electronic spectrum and states distribution of square Janus monolayer black arsenic phosphorus (b-AsP) quantum dots (QDs) in the presence of a perpendicular magnetic field are explored. Strong in-gap states of b-AsP QDs, whose probability densities are distributed on the armchair boundary (armchair edge states) appear in the energy gap of host perfect two-dimensional b-AsP. The corresponding energy levels of the armchair edge states can degenerate to the Landu energy levels upon applying a perpendicular magnetic field. When an in-plane polarized light is introduced, due to the presence of armchair edge states, the edge-to-edge transitions are mainly induced from the armchair edge (hole) states to zigzag edge (electron) states. The optical absorption undergoes blue shift as a function of the magnetic field. Our work suggests tunable optical properties via modulating the armchair edge states of a b-AsP QD and provides a theoretical basis for the design of b-AsP-based optoelectronic devices.