If $a,b,p,q,p',q'\in \mathbb{Z}$ satisfy $a(p^2+q^2)+bp'q'=a(p'^2+q'^2)+bpq$, is it true that $(p,q)=(q',p')$?

EDIT: This is wrong, I have made a new topic here: [Which linear transformations preserve this?](https://math.stackexchange.com/questions/1187519/which-linear-transformations-preserve-this)