$$ \frac{1}{2!}\langle c |\left(\{a^\dagger_j a_b\} \{a^\dagger_i a_a\} \hat{V}_N \right) | c\rangle = \frac{1}{2!}\langle c |\left( \hat{V}_N \{a^\dagger_a a_i\} \{a^\dagger_b a_j\} \right)^{\dagger} | c\rangle $$ which is nothing but $$ \frac{1}{2!}\langle c | \left( \hat{V}_N \{a^\dagger_a a_i\} \{a^\dagger_b a_j\} \right) | c\rangle^* =\frac{1}{2} \sum_{ijab} (\langle ij|\hat{v}|ab\rangle)^*\delta C_{ai}^*\delta C_{bj}^* $$ or $$ \frac{1}{2} \sum_{ijab} (\langle ab|\hat{v}|ij\rangle)\delta C_{ai}^*\delta C_{bj}^* $$ where we have used the relation $$ \langle a |\hat{A} | b\rangle = (\langle b |\hat{A}^{\dagger} | a\rangle)^* $$ due to the hermiticity of \( \hat{H} \) and \( \hat{V} \).