Moderating variables imply interaction effects, so that correlation between the independent and dependent variables is different between the subpopulations defined by the categories in the moderating variable..

If you have an interval-level moderating variable, the results can be harder to interpret, but the procedures are the same: first enter you independent variable and the moderating variable into a regression, and then run a second regression that also includes the interaction effect  (i.e., the product of the independent variable and the moderating variable) as a third variable.

There is no need for a moderator variable to be correlated with either of the others. In fact, here's a simple R demonstration, with fake data, of a dataset where none of the variables are correlated with each other but there is a significant interaction (significant moderation). You can run this code in R.

# Create normally distributed x data

x

Is there a reference to support this? I’m using Hays PROCESS and my results suggest there is an interaction but inital Perasons correlations showed no relationship between my moderator and predicitor.

There's no need for a reference because it's basic logic, and it's quite easy to make up fake examples where there's a legitimate interaction and the moderator has no correlation with the IV and DV.

I gave one such example above, here's another. Imagine you've got a sample of 400 people, such that 200 are university graduates and 200 are not (IV: education status). Within each group, 100 are French and 100 are Korean (moderator: nationality). And your DV can be whatever continuous variable you want it to be. Now imagine the results are like those shown in the attached graph: among the French, university graduates have a higher score on the DV than non-university graduates do, but among the Koreans the difference is the opposite; and there is no main effect of either the IV or the moderator. There you've got an interaction, and no possibility that the moderator is correlated with either the IV or the DV.

The collinearity (i.e. correlation) between X and Z should be avoided, for example by centering their values, i.e. subtracting their values from their mean. However, how it's possible to name a variable not associated to Y as a moderator?!! Can it moderate an association which itself is NOT related to any side of it?! Doesn't make sense!

Centering does not eliminate collinearity -- this is a widespread myth, but it should be obvious that subtracting a constant from a variable will not affect its correlation with the other variables in the model.

I agree with David Morgan's point of view.