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)Cu-4(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)Cu-4(PO4)(4) (A = Ba, Sr, and Pb) with the P42(1)2 space group. Despite their structural similarities, whereas the antiferromagnetic transition in A(TiO)Cu-4(PO4)(4) produces conventional anomalies in magnetization and heat capacity, that in K(NbO)Cu-4(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)Cu-4(PO4)(4) is far different from that of A(TiO)Cu-4(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)Cu-4(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)Cu-4(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.