Actually the symplectic structure is precious overkill right here, but let's stick to your sources. Let's keep the forms simple, so have mere pairs of variables, $\beta=(p,q)$, as @Kevin chose. Define $\gamma= (q',p')$.
A symplectic Fourier transform is just a double Fourier transform with a symplectic inner product (5,6) in the exponent, so, being cavalier about all normalizations,
$$
W(q',p')\propto \int dq dp ~~e^{i ~\gamma \wedge \beta} \operatorname{tr} ~(e^{-i \alpha \wedge \beta} |\psi\rangle \langle \psi|) .
$$
It is simple to work it out by application of elementary bra-ket algebra.
Note this is just a double plain Fourier transform of a simpler quantity, which most physicists use, namely
$$
W(q',p')\propto \int dq dp ~~e^{i ~(q'p-p'q) } \operatorname{tr} ~(e^{-i(\hat X p-\hat P q) } |\psi\rangle \langle \psi|) \\
\propto \int dq dp ~~e^{i ~(q'p+p'q) } \operatorname{tr} ~(e^{-i(\hat X p+\hat P q) } |\psi\rangle \langle \psi|)\\
\propto \int dq dp dz ~~e^{i ~(q'p+p'q) } \langle z|(e^{-i(\hat X p+\hat P q) } |\psi\rangle \langle \psi|z\rangle\\
\propto \int dq dp dz ~~e^{i ~(q'p+p'q) } e^{ipq/2} \langle z|(e^{-i\hat X p}e^{-i\hat P q } |\psi\rangle \overline{\psi}(z)\\
\propto \int dq dp dz ~~e^{i ~(q'p+p'q +pq/2 -pz) } \psi( z -q) \overline{\psi}(z)\\
\propto \int dq ~~e^{-ip'q} \psi(q'+q/2) \overline{\psi}(q'-q/2),
$$
where I've used the degenerate CBH identity for the Heisenberg algebra going to the 4th line, and performed the p integration to get a Dirac δ-function for z, which I then collapsed fortwith, and changed some variables.
Throughout, I've used $\hbar=1$ for simplicity. This expression for W is reviewed, e.g. (128, 139, 143) in our brief review A Concise Treatise on Quantum Mechanics in Phase Space, and was first essentially worked out in Moyal's epochal 1949 paper.
You could visualize this "poetically" as a glorified Dirac δ-function of sorts linking q ' to $\hat X$ and p ' to $\hat P$, so translating from traces in Hilbert space to phase-space functions.