= Solution
The first equation is vertical <hydrostatic equilibrium>: the pressure gradient balances the vertical gravity of the central mass, $\Omega^2z$, and the disk's own potential $\Phi_d$. The second is the plane-parallel <Poisson equation> for disk self-gravity. The third balances the vertical increase of <radiative flux> against local viscous heating in a Keplerian <alpha disk>, and the fourth is the optically thick <radiative diffusion> law.
With $P/\rho\sim c_s^2$, $\Sigma\sim\rho H$, and all terms in hydrostatic balance comparable,
$$
\frac{c_s^2}{H}\sim\Omega^2H\sim G\Sigma.
$$
The first comparison gives $H\sim c_s/\Omega$; inserting it into the second gives $\Omega c_s\sim G\Sigma$. Hence
$$
\boxed{Q=\frac{\Omega c_s}{\pi G\Sigma}\sim1},
$$
up to the order-one constants deliberately omitted by the scaling argument.
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