I think you want to show that in each ideal class the number of ideals of norm a is equal to the number of ideals of norm a*/D/. This should follow directly from the following 2 facts.

1. All primes p dividing /D/ ramify in the ring of integers of Q(root(D)).

2. The unique ideal of norm /D/ (uniqueness and existence follow from 1, taking a little care about the prime 2) is principal, generated by root (D).

In view of 1. and 2. the map J --> root(D) * J gives the desired bijection.

I think you want to show that in each ideal class the number of ideals of norm a is equal to the number of ideals of norm $a \cdot |D|$. This should follow directly from the following 2 facts.

(1). All primes $p$ dividing $|D|$ ramify in the ring of integers of $\mathbb{Q}(\sqrt{D})$.

(2). The unique ideal of norm $|D|$ (uniqueness and existence follow from (1), taking a little care about the prime $2$) is principal, generated by $\sqrt{D}$.

In view of (1) and (2) the map $J \rightarrow \sqrt{D} \cdot J$ gives the desired bijection.