  Let $(Z_t^{(q, H)})_{t \geq 0}$ denote a Hermite process of order $q \geq 1$
and self-similarity parameter $H \in (\frac{1}{2}, 1)$. Consider the
Hermite-driven moving average process $$X_t^{(q, H)} = \int_0^t x(t-u) dZ^{(q,
H)}(u), \qquad t \geq 0.$$ In the special case of $x(u) = e^{-\theta u}, \theta
> 0$, $X$ is the non-stationary Hermite Ornstein-Uhlenbeck process of order
$q$. Under suitable integrability conditions on the kernel $x$, we prove that
as $T \to \infty$, the normalized quadratic functional $$G_T^{(q,
H)}(t)=\frac{1}{T^{2H_0 - 1}}\int_0^{Tt}\Big(\big(X_s^{(q, H)}\big)^2 -
E\Big[\big(X_s^{(q, H)}\big)^2\Big]\Big) ds , \qquad t \geq 0,$$ where $H_0 = 1
+ (H-1)/q$, converges in the sense of finite-dimensional distribution to the
Rosenblatt process of parameter $H' = 1 + (2H-2)/q$, up to a multiplicative
constant, irrespective of self-similarity parameter whenever $q \geq 2$. In the
Gaussian case $(q=1)$, our result complements the study started by Nourdin
\textit{et al} in arXiv:1502.03369, where either central or non-central limit
theorems may arise depending on the value of self-similarity parameter. A
crucial key in our analysis is an extension of the connection between the
classical multiple Wiener-It\^{o} integral and the one with respect to a random
spectral measure (initiated by Taqqu (1979)), which may be independent of
interest.
