Let $(Z^{q, H}_t)_{t \in [0, 1]^d}$ denote a $d$-parameter Hermite random field of order $q \geq 1$ and self-similarity parameter $H = (H_1, \ldots, H_d) \in (\frac{1}{2}, 1)^d$. This process is $H$-self-similar, has stationary increments and exhibits long-range dependence. Particular examples include fractional Brownian motion ($q=1$, $d=1$), fractional Brownian sheet $(q=1, d \geq 2)$, Rosenblatt process ($q=2$, $d=1$) as well as Rosenblatt sheet $(q=2, d \geq 2)$. For any $q \geq 2, d\geq 1$ and $H \in (\frac{1}{2}, 1)^d$ we show in this paper that a proper normalization of the quadratic variation of $Z^{q, H}$ converges in $L^2(Ω)$ to a standard $d$-parameter Rosenblatt random variable with self-similarity index $H" = 1+ (2H-2)/q$.