(Transferred from PM conversation, hence the time stamps being so close together.)

Where did you use the definition of quadratic integer?

Quadratic integer - Wikipedia, the free encyclopedia

Note that D = -1 means we have and if a and b are integers then we have Gaussian integers for which all elements are quadratic integers

Gaussian integer - Wikipedia, the free encyclopedia

So in particular we know that a and b can't both be integer, at least one must be of the form p/q where q does not divide p. Furthermore if say b is non-integer but a is integer then a^2 + b^2 can't be integer, so a and b must both be non-integer.

So we're looking for a^2 + b^2 is an integer and a and b are both non-integers, subject also to the condition that a + bi must not be a quadratic integer. So while looking for a pair (a,b) to satisfy, it seems Pythagorean triples could help, for example a = 3/5, b = 4/5 where 3 and 4 are from a Pythagorean triple, with the result that a^2 + b^2 = 1. Now all we need to do is check that this is not a quadratic integer. So notice that for quadratic equation

both solutions are given by

Setting A = 1 as per definition of quadratic integer, we get

There is no way a number of this form can be

(notice the denominator 5 cannot be reduced to denominator 2), so we are done.