First, Mayans were not American. No, it is shown that ancient cultures had counting systems before there was a written language. For instance, the builders of Maccu Piccu used a system of strings with knots in them to represent units of 100. It helped them build terraces to get the rocks to the top of the mountain.
No, the Mayans were not the first to have a counting system and they are not Americans. Ancient civilization of the Olmec is believed to be the first to devise the counting system. This is because it is reported that they were discovered outside the Maya homeland and they adopted it soon after. Read more at http://en.wikipedia.org/wiki/Olmec#Writing.