From the answer of user1952009: https://math.stackexchange.com/q/2273035.
Looking for the general solution of system for prime $p$ and $q$: $$q^2+q+1\equiv 0 \bmod p\\ p^2+p+1\equiv 0 \bmod q$$ EDITED: The following solution found: $$p=3,q=13\\p=13,q=61\\p=22419767768701,q=107419560853453$$