Cosmic expansion has no influence on how the distance between the Sun and Alpha Centauri evolves. Not even a negligibly small influence, but none whatsoever.
The idea that cosmic expansion creates a tendency for things to separate is a common misconception, driven by an overly literal interpretation of the idea of expanding space. As I've said in another answer, space is not expanding, at least not in the way that you are thinking. I won't repeat all the details, so if you are in doubt, check that link. The basic point is that cosmic expansion is fundamentally just that galaxies and other material are moving apart, in the large-scale average.
With that in mind, this question is essentially asking how the recession of distant galaxies affects the distance between the Sun and Alpha Centauri. There is no effect, because the gravitational influence of galaxies in opposite directions cancel. This is the shell theorem in Newtonian gravity or Birkhoff's theorem in general relativity; in principle they only apply to spherically symmetric systems, but corrections due to asphericity introduce tidal forces, not any net outward pulls.
With that being said, people often conflate dark energy with cosmic expansion. Dark energy's gravitational influence accelerates cosmic expansion, but they are different things. If you are wondering instead whether dark energy influences the Sun-Alpha Centauri separation, the answer is likely yes, but not in the obvious way.
There is probably dark energy pervading the Milky Way. That dark energy would tend to push the Sun and Alpha Centauri apart. However, we can't talk about the Sun and Alpha Centauri in isolation, because their dynamics are dominated by their orbits around the Milky Way galaxy. In the context of galactic dynamics, dark energy effectively reduces the Galaxy's gravitational attraction. This would slightly perturb the Sun's and Alpha Centauri's orbits; they would move slightly more slowly, precess slightly differently, etc. But that's all. There's no tendency to make the stars move apart.
But there would be secondary effects on the Sun-Alpha Centauri separation due to their altered orbits. The two stars are only nearby today by chance; they were not close in the past and will not be in the future. Any minor perturbation to their orbits, such as that supplied by dark energy, could drastically change their positions within the Galaxy today. So in some sense, dark energy is responsible for the Sun and Alpha Centauri being near each other today!