  We report static and dynamic properties of the antiferromagnetic compound
Zn$_{2}$(VO)(PO$_{4}$)$_{2}$, and the consequences of non-magnetic Ti$^{4+}$
doping at the V$^{4+}$ site. $^{31}$P nuclear magnetic resonance (NMR) spectra
and spin-lattice relaxation rate ($1/T_1$) consistently show the formation of
the long-range antiferromagnetic order below $T_N= 3.8-3.9$\,K. The critical
exponent $\beta=0.33 \pm 0.02$ estimated from the temperature dependence of the
sublattice magnetization measured by $^{31}$P NMR at 9.4\,MHz is consistent
with universality classes of three-dimensional spin models. The isotropic and
axial hyperfine couplings between the $^{31}$P nuclei and V$^{4+}$ spins are
$A_{\rm hf}^{\rm iso} = (9221 \pm 100)$ Oe/$\mu_{\rm B}$ and $A_{\rm hf}^{\rm
ax} = (1010 \pm 50)$ Oe/$\mu_{\rm B}$, respectively. Magnetic susceptibility
data above 6.5\,K and heat capacity data above 4.5\,K are well described by
quantum Monte-Carlo simulations for the Heisenberg model on the square lattice
with $J\simeq 7.7$\,K. This value of $J$ is consistent with the values obtained
from the NMR shift, $1/T_1$ and electron spin resonance (ESR) intensity
analysis. Doping Zn$_2$VO(PO$_4)_2$ with non-magnetic Ti$^{4+}$ leads to a
marginal increase in the $J$ value and the overall dilution of the spin
lattice. In contrast to the recent \textit{ab initio} results, we find neither
evidence for the monoclinic structural distortion nor signatures of the
magnetic one-dimensionality for doped samples with up to 15\% of Ti$^{4+}$. The
N\'eel temperature $T_{\rm N}$ decreases linearly with increasing the amount of
the non-magnetic dopant.
