you need to show

reflexivity, i.e. aRa

symmetry, i.e. aRb <--> bRa

transitivity, i.e. (aRb,bRc) --> aRc

I don't see that restricting your points to Q makes any difference.

given that your relation is just the square of the distance between the points (x1,y1) and (x2,y2) the geometric interpretation of the equivalence classes should be pretty clear.