No. In his book Relativity: the special and the general theory he states quite clearly on page 134 of his Appendix IV:
"A few years later Hubble showed, by special investigation of extra-galactic nebulae that the spectral lines emitted showed a red shift which increased regularly with the distance of the nebulae. This can be interpreted in regard to our present knowledge only in the sense of Doppler's principle, as an expansive motion of the system of stars in the large as required, according to Friedman, by the field equations of gravitation. Hubble's discovery can, therefore, be considered to some extent as the confirmation of the theory"It is quite clear that Einstein was speaking about the general Doppler principle that the relative motions of a source and receiver of electromagnetic radiation are reflected in a spectral shift. Whether the relative separations are due to special relativistic motion in the usual sense of movement, or the dilation of space as is required by Friedman's cosmological solutions based on Einstein's own gravitational field equations, is not in dispute by Einstein.
The issue of what happens to the speed of light in all of this is that it ceases to be a GLOBAL constant of space-time because it only has meaning within local regions of essentially-flat spacetime where you can get away with using only special relativity to adequately describe relative motion. The moment your domain has grown so large that the cosmological effects of spacetime curvature and dilation become important, any statement about the speed of light being a globally-fixed constant in all such reference frames becomes inadequate. Special relativity is a local approximation to a far more complex global geometry for space-time.