I know that the color of light depends on its wavelength. Also, when light moves from one medium to another, its wavelength changes. The question: So, when light moves from one medium to another does its color change?
Another way of saying Apoorv Potnis's answer is to say that the light's color is defined by the photon energy, and this does not change. Since we always "see" light propagating through the vitreous humor of our eye, the only variable here is the frequency: the medium (vitreous humor) is defined and the wavelength follows. So, when we put our eye into water, or other medium, only the frequency can define the color.
There is another sense wherein the color can change subtly on refraction and that is if broadband (white) light is incident non-normally on an interface. Different wavelengths have different transmission co-efficients defined by the Fresnel Equations, so the frequency spectrum of the transmitted light is changed by the refraction. This effect is most pronounced at highly glancing incidence angles.
I'm giving a link of a similar question in my answer as I don't have enough reputation to comment. Link Does light color change when refracting?
The color we perceive is dependent on frequency and wavelength. The color will not change. What you're not taking into account is the speed of light in the medium. It's not the same $c$ in vacuum. The frequency stays the same. What changes is that speed of light in the refracting medium and as a result the wavelength. This difference for speed is the exact reason we have refractive effects, and I believe this was the observation that led to Snell's Law. In symbols
where $λ$ is the wavelength and $ν$ is the frequency.
The speed of light changes because the photons have to have its energy ( and therefore it's presence ) propagated across very long molecular chains. Electrons have to absorb the incident photons, re-emit and repeat this in a longitudinal direction. Depending on what the material is made of, this will take variable time in variable media. That notion manifests itself in the different indices of refraction that different objects have.
(just edited the answer to make it more complete, from other answers and comments)