First, we rewrite the Hamiltonian $$ \begin{align} H &=2\sum_{p}\delta_pa_p^{\dagger}a_p+\sum_{pq}g_{pq}a_p^{\dagger}a_q \tag{15}\\ &=\sum_{p}\left(2\delta_p+g_{pq}\right)a_p^{\dagger}a_p+\sum_{p\neq q}g_{pq}a_p^{\dagger}a_q. \tag{16} \end{align} $$ Applying the transformation to the first term in the Hamiltonian $$ \begin{align} a^{\dagger}_pa_p=\left(\frac{X_p-iY_p}{2}\right)\left(\frac{X_p+iY_p}{2}\right)=\frac{I_p-Z_p}{2}. \tag{17} \end{align} $$