Do any quantum algorithms improve on classical SAT?
Indeed, as wwjohnsmith1 said, you can get a square root speed-up over Schöning's algorithm for 3-SAT, but also more generally for Schöning's algorithm for k-SAT. In fact, many randomized algorithms for k-SAT can be implemented quadratically faster on a quantum computer.

The reason for this general phenomenon is the following. Many randomized algorithms for k-SAT that run in time $O(T(n) \mathrm{poly}(n))$ (where $T(n)$ is some exponentially growing function of $n$) actually do something stronger. At their core, there is a polynomial-time algorithm that outputs a satisfying assignment, if one exists, with probability at least $1/T(n)$. From this it is clear that if you repeat this poly-time algorithm $O(T(n))$ many times and accept if any of the runs returns a solution, you will get a randomized algorithm for k-SAT that runs in time $O(T(n) \mathrm{poly}(n))$.

Now instead of running this algorithm $O(T(n))$ times, you can run amplitude amplification on this poly-time algorithm. Amplitude amplification is a general quantum algorithm that can decide if another algorithm accepts with probability 0 or with probability $1/T$ using only $O(\sqrt{T})$ uses of this algorithm. Applying amplitude amplification to such a k-SAT solver will immediately yield a quantum algorithm for k-SAT with running time $O(\sqrt{T(n)}\mathrm{poly}(n))$, which is quadratically faster (ignoring the poly(n) term).