We give several criteria to show over which quadratic number fields
$\bQ(\sqrt{D})$ there should exists a non-constant arithmetic progressions of
five squares. This is done by translating the problem to determining when some
genus five curves C_D defined over Q have rational points, and then using a
Mordell-Weil sieve argument among others. Using a elliptic Chabauty-like
method, we prove that the only non-constant arithmetic progressions of five
squares over Q(sqrt{409}), up to equivalence, is 7^2, 13^2, 17^2, 409, 23^2.
Furthermore, we give an algorithm that allow to construct all the non-constant
arithmetic progressions of five squares over all quadratic fields. Finally, we
state several problems and conjectures related to this problem.