Abstract We report a combined In115 nuclear quadrupole resonance, V51 nuclear magnetic resonance, and muon spin-relaxation spectroscopic study of the low-temperature magnetic properties of InCu2/3V1/3O3, a quasi-two-dimensional (2D) compound comprising in the spin sector a honeycomb lattice of antiferromagnetically coupled spins S=1/2 associated with Cu2+ ions. Despite substantial experimental and theoretical efforts, the ground state of this material has not been ultimately identified.