This is just some of my thought: Suppose $\gamma = v_1/v_2$ is finite, we have $ x^T(A-\gamma B)y = 0$. Since $x,y$ are nonnegative and nonzero, at least the matrix $A-\gamma B$ cannot be positive for every entry. We can consider
$$f(x,y) \equiv x^T(A-\gamma B)y $$
as real-valued function defined on the compact space $T^{n-1} \times T^{n-1}$, where $T^{n}$ is the $n-$dimensional unit simplex. Now, the problem will be reduced to finding zeros of $f$ in this compact space. If we view $f$ as quadratic polynomial in with factors of $x_jy_k$, then maybe there are zero-finding numerical methods for this type of function.