Speed of light and speed of sound in the same medium

No direct relationship

Speed of light in medium depends only on material's electromagnetic properties: $$ v_{\textrm{light}}={\frac {1}{\sqrt {\mu \varepsilon }}} $$ Where as speed of sound in general depends on how pressure in material changes in relation to density change: $$ v_{\textrm{sound}}={\sqrt {\left({\frac {\partial p}{\partial \rho }}\right)_{s}}} $$

Indirect (weak) links

There is Clausius–Mossotti equation, which relates material permittivity to molecular polarizability $\alpha$ and number density of the molecules $N$ :

$$ {\frac {\varepsilon _{\mathrm {r} }-1}{\varepsilon _{\mathrm {r} }+2}}={\frac {N\alpha }{3\varepsilon _{0}}} $$

Number density relates to mass density in such way : $$ N={\frac {N_{\rm {A}}}{M}}\rho _{\mathrm {m} } $$ where $N_A$ is Avogadro constant, M - molar mass.

NO. There is no relation at all. Sound waves are mechanical waves whose speed is function of mechanical elastic properties of the medium, e.g. $v=\sqrt{\frac{E}{D}}$ in solids and $v=\sqrt{\frac{B}{D}}$ in gases and liquids. On the other hand light waves are e.m.w. whose speed is function of electric and magnetic properties of the medium or even vacuum given by $v=\sqrt{\frac{1}{\mu\epsilon}}$, where $E=$Young's modulus, $B=$Bulk modulus, $D=$density, $\mu=$magnetic permeability, $\epsilon=$electric permittivity.

If there IS a relationship between them, here is what it might look like.

The speed of light in air is set by the permeability and the permittivity of air, as expressed in the equation cited above by Agnius Vasiliauskas.

Those characteristics are set by the details of how electric charge is distributed around the gas molecules in the air, which determine how the molecules respond to changing electromagnetic fields in their vicinity.

Changes in the density of the air and/or its temperature will change the permeability and the permittivity of the air in bulk, and hence the speed of light going through it. Now note that changes in the density and the temperature of the air will also change the speed of sound going through it, as cited by Lionheart above.

The coupling that Tanmay seeks then looks like a relationship between the permittivity and permeability of air as functions of pressure and temperature, and the elasticity and mass density of air as functions of pressure and temperature.