I would guess this is to allow you to relate production to the other major environmental drivers such as light, temperature, nutrients etc which tend to follow a seasonal signal not necessarily strictly related to month but by climate (including ocean climate). So for instance relating to season is a better tool for capturing total annual production. Remember also that calender months are a slightly arbitary construct that can mast tidal cycles for instance.

My guess, best I can do based on freshwater macro invertebrates, is life cycles are short, some stages more difficult to sample and identify.  By being told to collect quarterly, someone has determined that frequency is probably sufficient, rather that the effort and cost it is to do monthly.  And some species populations may peak at different times, and the quarterly sampling may be sufficient to capture these population shifts and changes.  But if you have better information that would influence sampling frequency, you probably should bring it up.  

To optimize the environmental parameters with the population density

Favorable environmental conditions is vital for reproduction and larval settlements in most species take place in such conditions. So, their abundance may higher in such a good season than other. Some species occur only in one season, due to short life span. Such factors may affect your ecological studies. Studies in season will give you clear picture on their life cycle and life span.

The organisms do not "recognize" the human convention of months.  Their seasonal cycles and abundance cycles are cued to photoperiod (day length) and temperature, with other finer scale parameters influencing the onset of activity patterns and movements (migration, emigration, etc).  Seasonality may be an evolutionary means of partitioning resources (e.g.,  avoiding competition for food, space etc.) , or conversely tracking the abundance of resources through time and space.