On the space $S=\{ 0,1,\ldots,m \}^{\mathbb{N}}$ for some $m\in \mathbb{Z}_{+}.$ And given a probability $\mu$ on it. Is it true that $\mu$ is fully supported if and only if it is ergodic for the action of the shift?

No, there are many fullysupported nonergodic measures. Just take a convex combination of a fullysupported measure and anything else. (Recall that an invariant measure is ergodic if and only if it cannot be writen as a nontrivial convex combination of two distinct invariant measures.) Also there are many ergodic measures that are not fully supported. For example, given any periodic sequence consider the atomic measure that gives equal weight to each of the shifts of this sequence. This measure is ergodic but its support is a finite set. 

