With $$O_{\mu}^{\lambda} = \sum_{j_pj_q} \langle p \vert \vert O^{\lambda}\vert \vert q \rangle \lambda^{-1}\left[ a_{j_p}^{\dagger}\tilde{a}_{j_q}\right]^{\lambda}_{\mu},$$ we can write $$\langle \Phi^J_M|O^{\lambda}_{\mu}|\Phi^{J'}_{M'}\rangle = \sum_{pq} \langle p \vert O_{\mu}^{\lambda} \vert q \rangle \langle \Phi_{M}^{J} \vert a^{\dagger}_pa_q \vert \Phi_{M'}^{J'}\rangle,$$ as $$\langle \Phi^J_M|O^{\lambda}_{\mu}|\Phi^{J'}_{M'}\rangle = \sum_{j_pj_q} \langle p \vert \vert O^{\lambda} \vert\vert q \rangle \langle \Phi_{M}^{J}\vert \lambda^{-1}\left[ a_{j_p}^{\dagger}\tilde{a}_{j_q}\right]^{\lambda}_{\mu} \vert \Phi_{M'}^{J'}\rangle.$$ We have suppressed the summation over quantum numbers like $n_p,n_q$ etc.