In this work, the case of a Cox Process with Folded Normal Intensity (CP-FNI), in which the intensity is given by Lambda(t)=|Z(t)|, where Z(t) is a stationary Gaussian process, is studied. Here, two particular cases are dealt with: (i) when the process Z(t) constitutes a family of independent random variables and with a common probability law N(0,1), and (ii) the case in which Z(t) is a second order stationary process, with exponential type covariance function. In these cases, we observe that the properties of the Gaussian process Z(t) are naturally transferred to the intensity Lambda(t) and that very analytical results are achievable from the analytical point of view for the point process N(t). Finally, some simulations are presented in order to appreciate what type of counting phenomena can be modeled by these cases of CP-FNI. In particular, it is interesting to see how the trajectories show a tendency of the events to be grouped in certain periods of time, also leaving long periods of time without the occurrence of events.