I was trying to find in some general physics books and using the Internet the answer to my question - does each one of 8 gluons has its own name?
For example quarks are named:
But I am unable to find anything like this for gluons.
You're not finding anything, because they don't exist! The gluons don't really have names like the quarks do. But this is for a good reason - the obey (and mediate) an exact symmetry (that takes the form of SU(3)), which, in layman's terms, means that there is no way to distinguish colors except from how they interact with other colors, that we have already defined arbitrarily. So, there is no meaning to saying that "this gluon is this thing always, and this other gluon is this other thing always." On the other hand, quarks obey a nearly exact symmetry, SU(3) flavor symmetry. However, this is nearly exact, and not exact, which means it is broken to some extent! This is seen in the fact that the different flavors of quarks have different masses. Further, because the eigenstates of quarks that are most common in the world we interact with are mass eigenstates - the exact eigenstates for which the symmetry breaks down! - we choose to characterize the quarks (and name them) based on these eigenstates, which cannot simply be written in another, equally relevant basis.
So, it boils down to the fact that the representation of the gluons in a basis is arbitrary, and there is nothing else besides that arbitrary basis to distinguish them. Quarks, on the other hand, do have a "preferred" basis, so we use that as the convention for distinguishing the quarks.
