  We report a combined $^{115}$In NQR, $^{51}$V NMR and $\mu$SR spectroscopic
study of the low-temperature magnetic properties of InCu$_{2/3}$V$_{1/3}$O$_3$,
a quasi-two dimensional (2D) compound comprising in the spin sector a honeycomb
lattice of antiferromagnetically coupled spins $S=1/2$ associated with
Cu$^{2+}$ ions. Despite substantial experimental and theoretical efforts, the
ground state of this material was has not been ultimately identified. In
particular, two characteristic temperatures of about $\sim 40$ K and $\sim 20$
K manifesting themselves as anomalies in different magnetic measurements are
discussed controversially. A combined analysis of the experimental data
complemented with theoretical calculations of exchange constants enabled us to
identify below 39 K an ``intermediate'' quasi-2D static spin state. This spin
state is characterized by a staggered magnetization with a temperature
evolution that agrees with the predictions for the 2D XY model. We observe that
this state gradually transforms at 15 K into a fully developed 3D
antiferromagnetic N\'eel state. We ascribe such an extended quasi-2D static
regime to an effective magnetic decoupling of the honeycomb planes due to a
strong frustration of the interlayer exchange interactions which inhibits
long-range spin-spin correlations across the planes. Interestingly, we find
indications of the topological Berezinsky-Kosterlitz-Thouless transition in the
quasi-2D static state of the honeycomb spin-1/2 planes of
InCu$_{2/3}$V$_{1/3}$O$_3$.
