The characteristic function is 
$$\eqalign{ E\left[e^{itY}\right] &= E\left[ E\left[ e^{itY}|X \right]\right] \cr
&= E \left[ \exp(it\mu X - t^2 X^2/2 \right] \cr
&= \frac{\exp \left(\left(-(\alpha^2+\beta \mu^2) t^2 + 2 i \mu \alpha t\right)/\left(2 t^2 \beta + 2\right)
\right)}{\sqrt {{t}^{2}
\beta+1}}\cr}$$

It is certainly not a normal distribution, but might be approximated by a normal distribution when $\beta$ is small.