I assume that the equation you've given is of interest to you, so here is my interpretation of Sasha's comment. The equation under consideration has degree $2$ in all three variables, so a compactification of this surface in $\mathbb CP^1\times \mathbb CP^1\times \mathbb CP^1$ is a divisor of degree $(2,2,2)$. So if by any chance it is smooth, then indeed it is a K3 surface. It might happen, however that the surface has singularities in $\mathbb C^3$ or in one of $7=2^3-1$ other charts. So one needs to find the solution of $F_x=F_y=F_z=F=0$ in all these charts. Probably this can be done with a help of some program. Once these singularities are found one needs to check if these are Du Val or not (for this they need to be isolated of course). This should not be super hard since the degree in each variable is $\le 2$.