I need calculate the convolution
$$ Q(q, p) \equiv (\varphi_\vec{r}*W)(q, p) $$
Of a Wigner function $W(q, p)$ on real phase space with a multinormal distribution $\varphi_\vec{r}(q, p)$, where we have
$$ W(q,p)=\frac{1}{\sqrt{2 \pi} \,a^{1/3}} \, \mathrm{e}^{\frac{6 a p + 1}{12 a^2}} \, \mathrm{Ai} \left( \frac{a q^2 + p}{a^{1/3}} + \frac{1}{4 a^{4/3}} \right) $$
$$ \varphi_\vec{r}(q, p)= \frac{1}{\pi} \mathrm{e}^{-(q^2 + p^2)} $$
$$ \{q, p \} \in \mathbb{R} \; \mathrm{and} \; a>0 \,. $$
I was hoping this could be done using the convolution theorem
$$ (\varphi_\vec{r}*W)(q, p) = \mathcal{F}^{-1} \left( \mathcal{F}(\varphi_\vec{r}) \cdot \mathcal{F}(W) \right)(q, p) \; , $$
but I struggle to find the Fourier transform of $\mathcal{F}(W)$.
Context: $W(q, p)$ is the Wigner quasiprobability distribution of a non-Gaussian quantum state. I need its Husimi Q-function $Q(q, p)$, which is a physically equivalent quasiprobability distribution and can be calculated by convolution of the Wigner function with a Gaussian.