Since $\boldsymbol\Sigma$ is PSD, there exists $\boldsymbol Q$ s.t. $\boldsymbol Q^\top\boldsymbol Q=\boldsymbol \Sigma$ (by Cholesky, or eigendecomposition).
Then $\boldsymbol w^\top\boldsymbol \Sigma\boldsymbol w =\boldsymbol w^\top\boldsymbol Q^\top\boldsymbol Q\boldsymbol w = \|\boldsymbol Q\boldsymbol w\|_2^2.$
Then introduce new variable $y$ as epigraph of $\|\boldsymbol Q\boldsymbol w\|$ as follows:
\begin{align}y+\boldsymbol c^\top\boldsymbol w&=1\tag{linear constraint}\\\|\boldsymbol Q\boldsymbol w\|_2^2&\le y\tag{rotated quadratic cone constraint}\end{align}
As @mtanneau posted, these modeling tricks(?) are well-explained in the MOSEK tutorial.
If you use MOSEK, you can just directly use rotated quadratic cone to model second constraint.
If you want to convert into second order conic constraints,
\begin{align}&\|\boldsymbol{Qw}\|_{2}^{2} \leq y \\\iff&(\boldsymbol{Qw})^\intercal (\boldsymbol{Qw}) \leq y \\
\iff& (\boldsymbol{Qw})^\intercal (\boldsymbol{Qw})+\frac{(y-1)^2}{4} \leq \frac{(y+1)^2}{4}\\
\iff& \left\|\begin{pmatrix}\boldsymbol{Qw}\\ \frac{y-1}{2} \end{pmatrix}\right\|_{2} \leq \frac{y+1}{2}\end{align}
Hope this is what you're looking for.
*Converting quadratic constraint into soc constraint is well-known tricks, you can find in https://www2.isye.gatech.edu/~nemirovs/LMCOLN2021WithSol.pdf