No knowledge of Pell's equation is required when the desired solutions are rational numbers. Rearranging the original equation to the following: \begin{align*} x^2 - 1 & = dy^2\\ (x+1)(x-1)&= dy^2\\ \frac{x+1}{y} &= d\Big(\frac{y}{x-1}\Big) \end{align*} Setting parameter $t = \frac{y}{x-1} \in \mathbb{Q}$: $$\frac{x-1}{y} + \frac{2}{y} = dt \implies y = \frac{2t}{dt^2 - 1}$$ Accordingly $$x = \frac{dt^2+1}{dt^2-1}$$