A quadratic polynomial of $2n$ variables is given as $$ H = \sum_{i,j=1}^{2n} A_{ij} x_i x_j = x^T A x, $$ where $A$ is a symmetric matrix. I am looking for a symplectic transformation of these variables into $y = Cx$--i.e., $C^TJC=J$ where $J=\begin{pmatrix}0 & I\\ -I & 0\end{pmatrix}$--such that $H$ becomes diagonal in $y$'s: $C^TAC = D$ for some diagonal matrix $D$.
It is clear that an orthogonal transformation doing the job always exists, but the question is about symplectic transformations. In addition I think $D$ cannot be the Jordan normal form of $A$, since in that case $C$ can (must?) be orthogonal and $C^TC=I$ is generically in conflict with $C^TJC=J$.
The question arises naturally if you want to use canonical transformations of classical mechanics to convert the most general quadratic Hamiltonian of a set of coordinates and momenta into non-interacting harmonic oscillators.
In general, you cannot diagonalize a quadratic form on $\mathbb{R}^{2n}$ using a symplectic matrix. There is an analysis of all the possible canonical forms such a quadratic form might have, and it depends on the Jordan decomposition of the matrix $JA$. Check out Appendix 6 of Arnold's Mathematical Methods of Classical Mechanics for this analysis and a list of all possible normal forms of a quadratic Hamiltonian.