Puzzle 1214 pairs of primes (p & q) such that...

During a conversation about possible new puzzles Giorgos Kalogeropoulos and me came up with the following one, around March 23, 2025:

¿Are there pair of primes (p & q) such that:

a) q is a substring of p^2 and
b) p is a substring of q^2

The first example that Giorgos found was this one:

(p, q) = (3595601, 8346551)
3595601^2 = 12928346551201

8346551^2 = 69664913595601

Later he added the following three more examples:

p=8259499 and q=8219323731001

p=406348037 and q=1651187271737533

p=599963737 and q= 564857150051

Q. Can you find more examples?