  Static and dynamic properties of the quasi-two-dimensional antiferromagnet
K$_2$V$_3$O$_8$ have been investigated by $^{51}$V-NMR experiments on
nonmagnetic V$^{5+}$ sites. Above the structural transition temperature
$T_{\rm{S}}$ = 115 K, NMR spectra are fully compatible with the $P4bm$ space
group symmetry. The formation of superstructure below $T_{\rm{S}}$ causes
splitting of the NMR lines, which get broadened at lower temperatures so that
individual peaks are not well resolved. Evolution of NMR spectra with magnetic
field along $c$-axis below the magnetic transition temperature $T_{\rm{N}} \sim
4$ K is qualitatively consistent with a simple N\'{e}el order and a spin flop
transition. However, broad feature of the spectra does not rule out possible
incommensurate spin structure. The spin-lattice relaxation rate $1/T_1$ below
$T_{\rm{N}}$ shows huge enhancement for a certain range of magnetic field,
which is independent of temperature and attributed to cross relaxation due to
anomalously large nuclear spin-spin coupling between V$^{5+}$ and magnetic
V$^{4+}$ sites. The results indicate strong gapless spin fluctuations, which
could arise from incommesurate orders or complex spin textures.
