Quasi-zero-dimensional antiferromagnets with weakly coupled clusters of multiple spins can provide an excellent platform for exploring exotic quantum states of matter. Here, we report the synthesis and the characterization of a copper-based insulating antiferromagnet, K(NbO)Cu4(PO4)4. Single-crystal X-ray diffraction measurements reveal that the crystal structure belongs to the tetragonal space group P4/nmm, in which Cu2+ ions align to form a quasi-two-dimensional layer of spin-1/2 coupled square tetramers. The structure is quasi-isostructural to recently reported magnetoelectric antiferromagnets, A(TiO)Cu4(PO4)4 (A = Ba, Sr, and Pb) with the P4212 space group. Despite their structural similarities, whereas the antiferromagnetic transition in A(TiO)Cu4(PO4)4 produces conventional anomalies in magnetization and heat capacity, that in K(NbO)Cu4(PO4)4 has several unusual features such as an upturn in magnetic susceptibility and a very weak specific heat anomaly that corresponds to a spin entropy release as small as 3%. These results indicate that the magnetism of K(NbO)Cu4(PO4)4 is far different from that of A(TiO)Cu4(PO4)4 and suggest that the ground state is very close to a quantum nonmagnetic singlet state. The origin of the distinct magnetism in K(NbO)Cu4(PO4)4 is discussed in terms of structural modifications of a Cu4O12 unit forming a square tetramer. Our study demonstrates that the present material family, represented by an extended chemical formula A(BO)Cu4(PO4)4 (AB = KNb, BaTi, SrTi, and PbTi), has broad chemical controllability of their magnetism. This makes this system an attractive material platform to study the physics of quantum spin-1/2 coupled square tetramers.