Write out the definition of the expectation:
$$
\mathbb{E}\left(\exp(X^2)\right) = \int_{-\infty}^{\infty} \mathrm{e}^{x^2} \frac{1}{\sqrt{2\pi} \sigma} \exp\left(-\frac{1}{2} \frac{x^2}{\sigma^2} \right) \mathrm{d} x
$$
When does this integral converge?
As to the distribution law of $Y=\exp(X^2)$, assuming $y>1$
$$
F_{Y}(y) = \mathbb{P}\left(\exp(X^2) < y \right) = \mathbb{P}\left(X^2 < \log(y)\right) = F_{X^2}(\log(y))
$$
Therefore the probability density function of $Y$:
$$
f_Y(y) = F_{Y}^\prime(y) = \frac{1}{y} f_{X^2}(\log(y)) = \frac{1}{\sqrt{2 \pi } \sigma ^2 \sqrt{\log (y)}} y^{-\frac{1}{2 \sigma ^4}-1}
$$