Not necessarily. Sometimes the term "z value" is used to describe a standardized variable $Z$ (or a single value $z$), which is a set of observed values (realizations of some distribution) with a population mean of exactly 0, and a population standard deviation of exactly 1 derived from another variable $X$ through the transformation:
$$Z = \frac{X-\mu_{X}}{\sigma_{X}}$$
If the distribution of $X$ is normal, then $Z$ will have a standard normal distribution. However, if $X$ is not normal, then $Z$ will not be either, even though it still has a population mean of exactly 0, and population standard deviation of exactly 1 (i.e. $\mu_{Z} = 0, \sigma_{Z}=1$).