What is the relation between input and output PSDs given system transfer function $H(s)$
If the system described by the transfer function $H(s)$ is stable, you can obtain its frequency response by substituting $s=j\omega$, and use the relation that you found:
$$S_Y(\omega)=S_X(\omega)\big|H(j\omega)\big|^2\tag{1}$$
where $S_X(\omega)$ and $S_Y(\omega)$ denote the power spectra of the system's input and its output, respectively.