for $X \sim \mathcal{N}(0,I_d)$,
\begin{align*}
\E[\norm{\Sigma^{\frac{1}{2}}X}^2] &= \int_{\R^d} \norm{\Sigma^{\frac{1}{2}}x}^2\cdot \frac{1}{(2 \pi)^\frac{d}{2}} e^{-\frac{1}{2}x^Tx}\,\d x \\ &= \int_{\R^d} x^T \cdot \Sigma \cdot x\cdot \frac{1}{(2 \pi)^\frac{d}{2}} e^{-\frac{1}{2}x^Tx}\,\d x \\ &= \sum_{i,j} \sigma_{i,j} \int_{\R^d} x_ix_j \frac{1}{(2 \pi)^\frac{d}{2}} e^{-\frac{1}{2}x^Tx}\,\d x = \sigma_{1,1} + ... +\sigma_{d,d}, \end{align*}
this should be it if I don't overlook something ovious