Magnetic properties of AMoOPO4Cl (A=K,Rb) with Mo5+ ions in the 4d1 electronic configuration are investigated by magnetization, heat capacity, and nuclear magnetic resonance (NMR) measurements on single crystals, combined with powder neutron diffraction experiments. The magnetization measurements reveal that they are good model compounds for the spin-1/2 J1−J2 square-lattice magnet with the first and second nearest-neighbor interactions. Magnetic transitions are observed at around 6 and 8 K in the K and Rb compounds, respectively. In contrast to the normal Néel-type antiferromagnetic order, the NMR and neutron diffraction experiments find a columnar antiferromagnetic order for each compound, which is stabilized by a dominant antiferromagnetic J2. Both compounds realize the unusual case of two interpenetrating J2 square lattices weakly coupled to each other by J1.