I've recently come across the term symplectic Fourier transform (see this paper, first page, second column), but googling didn't lead me to any satisfactory explanation of what is meant with this term.
The wikipedia page on the Fourier transform does refer to symplectic stuff in a couple of places, but I find most references there quite cryptic.
What is the symplectic Fourier transform?
To add some context, in the paper linked above the authors mention that the Wigner function $W_\psi(p,q)$ defined as
$$W_\psi(p,q) = \pi^{-1}\int_{\xi\in\mathbb R}e^{-2\pi i\xi p}\overline{\psi}\left(q-\frac{1}{2}\xi\right)\psi\left(q+\frac{1}{2}\xi\right)$$
is the symplectic Fourier transform of the characteristic function $\Xi_\psi$ defined as (I'm using the braket notation common in quantum information theory here):
$$\Xi_\psi(p,q)=\operatorname{tr}\left(w(p,q)^\dagger \lvert\psi\rangle\!\langle\psi\rvert\right)$$
where $w(p,q)\equiv e^{i(p\hat X-q\hat P)}$.
Best Answer
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.