Can palm fronds that fall into a swimming pool turn the pool water green?

Left to decay in the water it can release the chlorophyll (which is green) as well as other plant cell material which can feed an algae problem. Anything that goes into the water has some contaminants on them which will contribute to an algae problem. If it is left in the water to decay (already well on its way by the time it sinks), the insides of the plant cells are spilled out into the water and have to be counteracted with chlorine (either by shock, tablets, chlorine generators, etc.). Neglect is the root cause of most algae problems. If the pool was properly maintained that (algae) should not occur.
Yes, and to add to that, the added stress to the filtration system to go beyond it's ability to remove those contaminants can and will become major problems in the near future. It is incomprehensible to think you would even consider leaving debris in the pool. Left to go out of control in that manner is an unnecessary expense and detrimental to the surface of the pool and equipment.