Primes as quotients

I ask this question based on a comment of David Speyer in another question. What primes are of the form
$$
\frac{p^2-1}{q^2-1}
$$
where $p$ and $q$ are prime?

The first prime not apparently of this form is 17. The Diophantine equation
$$
p^2-17q^2+16=0
$$
has solutions following a linear recurrence relation which has no primes in the first 1000 terms (only $(\pm1, 1)$ seeds may contain primes). But perhaps there is a better way to go about this?

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