How is distributed the sum of square of correlated z-score?

I have a $1\times k$ matrix representing $z$-scores and each element is correlated to each other according to a covariance matrix $\Sigma$. I would like to compute their sum of square and to know the resulting distribution, but since values are correlated, it may not be $\chi^2$ distributed.

I have found this link : sum of squares of dependent gaussian random variables, but I have difficulty to understand and implement the solution.