No; the galaxies are just sitting there minding their business; it's the expansion of the intervening space (between the distant galaxies and us) that creates the sense of movement.
Think of it this way; if no force acts on the mass in the galaxies, it's velocity hasn't changed, and thus there are no associated Special Relativistic effects, e.g. the effective mass of the galaxy doesn't get larger.
The relativistic effects that we DO see (redshift) are, as the article points out, due to General, not Special, Relativity, and are a consequence of the spatial expansion.
Grumble, grumble ... maybe, but I donno. God not only doesn't play dice, he doesn't toss Frisbees either.