A generalized chi-squared distributed random variable $\xi$ can be expressed as $$\xi=q(x)=x'Q_2x+q_1'x+q_0,$$ where $x\sim\mathcal{N}\left(\mu, \Sigma\right)$, $Q_2$ is a matrix, $q_1$ is a vector and $q_0$ is a scalar.
Question: Would a multivariate version of this distribution have the form $$\left(\bigoplus_{i=1}^n x'Q_2(i)x\right)i_n + Q_1\left(\bigoplus_{i=1}^p x_i\right)i_p + \vec{q_0}\,\,? \,\,\tag{$1$}$$ where $x'=(x_1,\ldots,x_p)\sim\mathcal{N}(\mu, \Sigma)$ and $Q_2(i)$ depends on $i$, and $i_p$ is a $p\times $1 vector of all $1$'s.
To ensure that each of the three terms in the sum above is an $n\times 1$ vector, $Q_1$ is an $n\times p$ matrix and $Q_2$ is a $p\times p$ matrix.
Background: Each component of my vector is a non-central chi-square variable. I started by writing a vector, and then made the vector expression look as close to the univariate expression as possible (so that I could guess its distribution). In the univariate case, I had, for the $i$th entry of the vector, $$\sum_{k=1}^p\sum_{j=1}^p\sum_{l=1}^n a_{l,k}b_{i,j,l}x_{j}x_{k}+\sum_{j=1}^n a_{i,j}x_j+q_0,$$ and then I used algebra to rewrite the entire vector in the form $\left(1\right)$.