What is an example where Pearson correlation coefficient between 2 variables might be zero even when they have a relationship between them ?
The Pearson correlation coefficient only measures linear relationship. So, if the Pearson correlation coefficient is 0, it does not mean that there is no relation between two variables rather it means that there is no linear correlation between the variables.
Consider the below image:
Here, there is a clear non-linear relationship between the variables. So, in this case the Pearson correlation will be close to zero but the variables are not uncorrelated.