If not, are there some relations between them?
Not sure about the condensed matter context, but in general the answer is NO. For instance, quarks have fractional charge but are regular fermions.
The answer is 'no' in condensed matter also. As anyons are neither bosons nor fermions, they can follow some statistics other than BE or FD, but it has nothing to do with fractional charge. just-learning has already given you a perfect example.