hard or easy ? more, please...

[Maximilian Hasler]

Dear SeqFans,

I look for pairs of primes {p,q} such that
(p+1)(p+2)=-1 (mod q)
(q+1)(q+2)=-1 (mod p)

but I lack inspiration.

It appears as if the first congruence has only solutions if q=1 mod 6
(?), so this must be true for both, p and q.
I only found {7,73} and {8647, 94321}, no other below 10^6 (not
astonishing if growth continues like for the first 2 terms...).
Most OEIS hits for these are obviously "false positive" (concatenation
stuff and such),
searching "7 8647" gives some maybe related sequences (primes of form n^2-2 ?)
but I'm clueless for now...

Sorry for posting such uninteresting and/or trivial stuff, but if you
happen to know the solution or any reference, thanks in advance.
Maximilian