Let $A$ be a symmetric matrix and $X = (X_1,...,X_n)^T$ a vector of independent identically distributed random variables. The random variables are assumed not to be normally distributed.
What ist the variance of the quadratic form $x^T A x$, i.e.
$Var(x^T A x) = ?$
Writing $X^T A X = \sum_{i,j} A_{ij} X_i X_j$, we have $$ \text{Var}(X^T A X) = \sum_{i,j,k,l} A_{ij} A_{kl} \text{Cov}(X_i X_j, X_k X_l)$$ The covariance is $0$ if $\{i,j\}$ and $\{k,l\}$ are disjoint, but all the other terms are potentially nonzero. You'll have to consider various cases. Without knowing anything about the distribution of the $X_i$, there's not much more that can be said.