We study the zero-temperature phase diagram of the J1 XXZ-J2 XXZ Heisenberg model for spin-1 particles on an infinite square lattice interacting via nearest-neighbour () and next-nearest-neighbour (J2>0) bonds. The two bonds have the same XXZ-type anisotropy in spin space. The effects on the quasiclassical Néel-ordered and collinear stripe-ordered states of varying the anisotropy parameter Δ are investigated using the coupled cluster method carried out up to high orders. By contrast with the case for spin- particles studied previously, no intermediate disordered phase between the Néel and collinear stripe phases, for any value of the frustration J2/J 1, for either the z-aligned (Δ>1) or xy-planar-aligned (0≤Δ<1) states, is predicted here. The quantum phase transition is determined as first order for all values of J2/J1 and Δ. The position of the phase boundary J2 c(Δ) is determined accurately. It is observed to deviate most from its classical position (for all values of Δ>0) at the Heisenberg isotropic point (Δ = 1), where J2 c(1) = 0.55 ± 0.01. By contrast, at the XY isotropic point (Δ = 0), we find J2 c(0) = 0.50 ± 0.01. In the Ising limit (), as expected.