Can one star be observed at different wave length show different redshift ?
It's not impossible, but does require a bit if a special set-up.
Normally, the light from a star is redshifted mainly due to its velocity toward/away from us (a small, but important, contribution comes from gravitational redshift, when the light climbs up the potential of the star). Because stars tend to rotate, light coming from the side of the star rotating away from us is slightly more redshifted than the light coming from the side rotating toward us. If a large prominence is present on one side, then because the prominence is hotter than the average surface temperature of the star and thus emits at shorter wavelengths, the combined spectrum will exhibit a larger redshift at short wavelengths. Since the flux from the prominence alone is, after all, small compared to the total flux, the effect would be rather small, however.
A more common mechanism that may cause a differential redshift is stellar outflows. If large amounts of gas are being expelled (e.g. in protostars or AGB stars), then atoms and molecules in the outflowing material will absorb some of the light from the star. These absorption lines are then blue shifted wrt. the "systemic" velocity (i.e. the velocity of the star wrt. us), again skewing the redshift as a function of wavelength. A spectral line may even be seen both in emission and absorption, known as a so-called P-Cygni profile. The figure below (from the Wiki article linked to) shows the same intrinsic wavelength (here H$\alpha$) both redshifted and blueshifted (by rotation and thermal motion), but the blueshifted light is absorbed by the outflowing atmosphere:
If I am not mistaken, when analyzing the spectrum of a star, in order to measure its redshift, we need to find several well known absorption lines (which occur at different wavelengths) shifted by the same amount: the redshift (actually we also need to characterize as well as possible the star too). It follows that you do not get different redshifts.
However, when looking at a nearby object at different wavelengths, the sun for example, we may be looking at different parts of it. These may be moving toward or away from us. There, you would see a redshift for parts moving away from us, but this is just the doppler effect from a moving object, nothing to do with the universe expanding.