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 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.