I am just writing as an answer the point raised by Alex Degtyarev in the comments: for every root $a_i$ of $P(x)$, resp. for every root $b_j$ of $Q(y)$, the surface contains the rational curve $\{(a_i,y,0):y\in \mathbb{C}\}$, resp. the rational curve $\{(x,b_j,0):x\in \mathbb{C}\}$.

Edit. The OP clarifies that the question is whether the Zariski open subset $D(y)$ is hyperbolic. This open subset is a finite, unbranched cover of the affine hyperbolic surfaces $$[\mathbb{C}\setminus Z(P(x))]\times [\mathbb{C}\setminus Z(Q(y))],$$ the product of two affine hyperbolic curves.

I am just writing as an answer the point raised by Alex Degtyarev in the comments: for every root $a_i$ of $P(x)$, resp. for every root $b_j$ of $Q(y)$, the surface contains the rational curve $\{(a_i,y,0):y\in \mathbb{C}\}$, resp. the rational curve $\{(x,b_j,0):x\in \mathbb{C}\}$.

Edit. The OP clarifies that the question is whether the Zariski open subset $D(z)$ is hyperbolic. This open subset is a finite, unbranched cover of the affine hyperbolic surfaces $$[\mathbb{C}\setminus Z(P(x))]\times [\mathbb{C}\setminus Z(Q(y))],$$ the product of two affine hyperbolic curves.

I am just writing as an answer the point raised by Alex Degtyarev in the comments: for every root $a_i$ of $P(x)$, resp. for every root $b_j$ of $Q(y)$, the surface contains the rational curve $\{(a_i,y,0):y\in \mathbb{C}\}$, resp. the rational curve $\{(x,b_j,0):x\in \mathbb{C}\}$.

I am just writing as an answer the point raised by Alex Degtyarev in the comments: for every root $a_i$ of $P(x)$, resp. for every root $b_j$ of $Q(y)$, the surface contains the rational curve $\{(a_i,y,0):y\in \mathbb{C}\}$, resp. the rational curve $\{(x,b_j,0):x\in \mathbb{C}\}$.

Edit. The OP clarifies that the question is whether the Zariski open subset $D(y)$ is hyperbolic. This open subset is a finite, unbranched cover of the affine hyperbolic surfaces $$[\mathbb{C}\setminus Z(P(x))]\times [\mathbb{C}\setminus Z(Q(y))],$$ the product of two affine hyperbolic curves.