Let $a,b\in\mathbb Z$ be squarefree with $a>0$. Suppose that I know that there exist $(0,0,0)\neq (x,y,z) \in \mathbb Z^3$ s.t. $x^2-by^2-az^2=0$. Is there any known algorithm to find any such a triple?
Hint: Since your equation is homogeneous, an integer solution can be interpreted as a rational point on the corresponding projective curve. Note that this curve has genus $0$.
In particular, you can use Magma to solve your problem, here is an example.
Try this code in the magma calculator (http://magma.maths.usyd.edu.au/calc/) and find that $(50,9,5)$ is an integer solution to $x^2-25y^5-19z^2=0.$