= Solution
For a radial <null geodesic> in a spatially flat universe, the <comoving radial distance> increment is $d\chi=c\,dz/H(z)$. If the cloud and <quasar> redshifts differ only slightly, take $H$ at their common approximate epoch $z_*$, giving
$$
\Delta\chi\simeq\frac{c\,\delta z}{H(z_*)},\qquad H(z_*)=H_0\sqrt{\Omega_{m,0}(1+z_*)^3+\Omega_{\Lambda,0}}.
$$
Multiplying by the <scale factor> at that epoch gives the local <proper distance in cosmology> relevant to the quasar's radiation flux:
$$
\boxed{d_{cq}\simeq\frac{c\,\delta z}{H_0(1+z_*)\sqrt{\Omega_{m,0}(1+z_*)^3+\Omega_{\Lambda,0}}}.}
$$
If distance is instead expressed in comoving units, it is $(1+z_*)d_{cq}$. Equivalently the small rest-frame velocity difference is $\delta v=c\,\delta z/(1+z_*)$, and $d_{cq}\simeq\delta v/H(z_*)$. This <quasar-cloud redshift separation> assumes the redshift difference is due to homogeneous expansion, with negligible <peculiar velocities>. Since the two observations lie on a light cone, the proper separation is understood as the short separation on a nearly common cosmic-time slice; specifying the epoch matters at higher order.
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