  We study the zero-temperature phase diagram of the
$J_{1}^{XXZ}$--$J_{2}^{XXZ}$ Heisenberg model for spin-1 particles on an
infinite square lattice interacting via nearest-neighbour ($J_1 \equiv 1$) and
next-nearest-neighbour ($J_2 > 0$) bonds. Both bonds have the same $XXZ$-type
anisotropy in spin space. The effects on the quasiclassical N\'{e}el-ordered
and collinear stripe-ordered states of varying the anisotropy parameter
$\Delta$ is investigated using the coupled cluster method carried out to high
orders. By contrast with the spin-1/2 case studied previously, we predict no
intermediate disordered phase between the N\'{e}el and collinear stripe phases,
for any value of the frustration $J_2/J_1$, for either the $z$-aligned ($\Delta
> 1$) or $xy$-planar-aligned ($0 \leq \Delta < 1$) states. The quantum phase
transition is determined to be first-order for all values of $J_2/J_1$ and
$\Delta$. The position of the phase boundary $J_{2}^{c}(\Delta)$ is determined
accurately. It is observed to deviate most from its classical position $J_2^c =
{1/2}$ (for all values of $\Delta > 0$) at the Heisenberg isotropic point
($\Delta = 1$), where $J_{2}^{c}(1) = 0.55 \pm 0.01$. By contrast, at the XY
isotropic point ($\Delta = 0$), we find $J_{2}^{c}(0) = 0.50 \pm 0.01$. In the
Ising limit ($\Delta \to \infty$) $J_2^c \to 0.5$ as expected.
