Hi,
#1 Take the right-hand road.
Can I say "right" instead of "right-hand"? Yes.
#2 Take the right road.
If so, does #2 mean the same as #1? Yes.
#1 would be more common than #2. More common still, I think, would be Take the road to/on the right. Or other expressions, like Go right.
#2 seems a little odd, probably because it could also mean 'the correct road'. I think that's probably why it's less common.
Best wishes, Clive