Nitrogen (nitrate works, though ammonia is usually more easily utilised) is certainly required for growth.
Having no detectable nitrate/nitrite/ammonia in the aquarium theoretically could limit growth...but in order for that to happen, all other requirements would have to be met.
It's far more likely that another trace element, micro-nutrient or macro-nutrient (or other environmental condition - anything from an excess of growth-inhibiting compounds to not enough food/light/etc) is the limiting factor for your growth, rather than not enough nitrogen.
That said, it would be pretty easy to check - lots of companies make nitrogen supplements for aquarium use. I like Seachem's Flourish Nitrogen, personally, but you could use any of them (in conjunction with a good set of testing equipment!) to experiment.
Here is a must read for all serious aquarists. It may not aways be an easy read but if you can get through it you will have a much better understanding of some of the myths that continue to be perpetuated in this hobby. It may help you to adopt a different (and more accurate) way of thinking about coral classifications and distinctions as it relates to coral's requirements as it relates to health, growth, coloration etc. . It will very directly address the question posed here. See what you think.
I maintenence a doctors tank and he will not pay to do it right, so we go longer than we should on water changes and additives. He mainly has softies and a few hard corals. The corals he has grows twice as fast as mine do. I have actually put stuff in there so I could speed up growth. Thanks!!