expected value of two LTI output signals multiplied (cross correlation)
Assumimg that the two linear systems are BIBO-stable, the random processes $\{Y_1(t)\}$ and $\{Y_2(t)\}$ are zero-mean WSS Gaussian processes with autocorrelation functions and power spectral densities given by 
\begin{align}
R_{Y_1} &amp;= \sigma^2 (h_1 \star \tilde{h}_1)\\
R_{Y_2} &amp;= \sigma^2 (h_2 \star \tilde{h}_2)\\
S_{Y_1} &amp;= \sigma^2 |H_1|^2\\
S_{Y_2} &amp;= \sigma^2 |H_2|^2
\end{align}
In fact, the processes are also jointly Gaussian and jointly WSS processes with cross-correlation function $$R_{Y_1, Y_2}(\tau) = E[Y_1(t), Y_2(t+\tau)] = \sigma^2 (h_1 \star \tilde{h}_2)$$ and cross-power spectral density 
$$S_{Y_1,Y_2}(f) = \sigma^2 H_1(f)H_2^*(f).$$
The OP wants to find $E[Y_1(t)Y_2(t)]$ which is given by
\begin{align}E[Y_1(t)Y_2(t)] &amp;= R_{Y_1,Y_2}(0)\\
&amp;= \sigma^2 h_1\star \tilde{h}_2\big|_{\tau=0}\tag{1}\\
&amp;= \sigma^2\int_{-\infty}^{\infty} H_1(f)H_2^*(f) \,\mathrm df\tag{2}
\end{align}
since the OP prefers the frequency-domain calculation. Personally, given only $h_1$ and $h_2$ (and not $H_1$ and $H_2$), I would say that it is easier/cheaper to just grind out $(1)$ rather than use the frequency-domain calculation $(2)$ but the OP might have specific reasons for opting for the frequency-domain calculation. In particular, $\operatorname{var}(Y_1+Y_2)$ which is what the OP seems to really want to find is just
$$\operatorname{var}(Y_1+Y_2) = \sigma^2\begin{cases}\displaystyle\int_{-\infty}^\infty |h_1(t)+h_2(t)|^2 \,\mathrm dt,\\
\displaystyle\sum_{n=-\infty}^\infty |h_1[n]+h_2[n]|^2\end{cases}$$
which seems easier than the frequency-domain version of the same calculation, but ymmv.