Here is an infinite family of solutions resulting from setting
$x=A z$ for integer $A$.

For example set $x=2z$ and get 

$$g(y,z)=-(y^2 - 8*z^2 - z)*z$$

The quadratic factor is conic and
[Wolfram Alpha gives](https://www.wolframalpha.com/input?i=solve+-y%5E2*z+%2B+8*z%5E3+%2B+z%5E2+++over+integers) infinitely many integer
solutions in terms of powers of square root of two,
e.g: $f(2*36,102,36)=0$

Potential attack might be to try rational $A$ and then find integral
points on a conic with rational coefficients.