In the study of the Diophantine Equation $f(x)f(y) = f(z^2)$ where $f$ is quadratic, the computational proofs I have seen (for specific $f$) rely on Pell's Equation.
For example, if $f(t) = t^2+t+1$, then the equation becomes
$$(x^2+x+1)(y^2+y+1) = (z^4+z^2+1)$$
and via the substitution $y=x+2z$, and one more subsequent substitution, one eventually reaches the generalised Pell Equation
which is known to have infinitely many positive integer solutions.
Has anyone come across other methods of showing infinitely many (nontrivial) integer or rational solutions for quadratic $f$ that do not rely on Pell Equations?