The DFT is well-developed over $\mathbb{C}$ with fast quantum algorithms.
There is a DFT defined classically over $F_q$ which mirrors the complex case when we have an $N^{th}$ root of unity in $F_q$, i.e. $N|q-1$.
In a sense, I suppose we must work over $\mathbb{C}$ on quantum computers, since the amplitudes are naturally complex. Nevertheless, we can write $x \in F_q$ as $|x\rangle \in \mathcal{H}$ where we use $\lfloor \log_2 q \rfloor + 1$ qubits.
The only definition I could find is de Beaudrap, Cleve, and Watrous, $\S 2.1$:
Let $\phi : GF(q) \rightarrow GF(p)$ be any nonzero linear mapping. The QFT with respect to $\phi$ is
$$F_{q,\phi}: |x \rangle \mapsto \frac{1}{\sqrt{q}} \sum_{y \in GF(q)} \omega^{\phi(xy)}|y \rangle $$
for $\omega = e^{2 \pi i / p}$ and extend $F_{q,\phi}$ by linearity.
Is this the only definition of a QFT over a finite field?
