Expectation of product of Filtered White Gaussian Noise
In general, for an input signal $x$ with power-spectrum density $S_x(f)$ passed through a linear-time-invariant (LTI) system with transfer function $H(f)$, the power-spectrum density of the output, $S_y(f)$ is given by:

$$
  \begin{align}
    S_y(f) &amp;= S_x(f)\left|H(f)\right|^2
  \end{align}
$$
A complete derivation can be found in various textbooks (such as "Communication Systems Engineering", by John G. Proakis and Masoud Salehi) and lecture notes (such as these). Note that a consequence of this result for Gaussian noise is also quoted here.

Thus, if evaluating the output PSD at $f_1$ and $f_2$, we have
$$
  \begin{align}
    S_y(f_1) &amp;= S_x(f_1)\left|H(f_1)\right|^2 \\
    S_y(f_2) &amp;= S_x(f_2)\left|H(f_2)\right|^2
  \end{align}
$$
And consequently,
$$
  \begin{align}
    E[S_y(f_1)S_y(f_2)]
      &amp;= S_y(f_1)S_y(f_2) \\
      &amp;= S_x(f_1)S_x(f_2)\left|H(f_1)H(f_2)\right|^2
  \end{align}
$$

Note that for an input consisting of white (or at least white over some finite observable bandwidth much greater than B) Gaussian noise with power-spectral density $N_0/2$, we get:
$$
  \begin{align}
    E[S_y(f_1)S_y(f_2)] &amp;= \frac{1}{4} N_0^2\left|H(f_1)H(f_2)\right|^2
  \end{align}
$$