Children do not see more colors than adults.
The cone cell of the human eye is the main type of photoreceptor that senses differences in light bandwidth, aka color. The dics of the cone cell are tuned to be receptive to certain bandwidths of light. These bandwidths are well understood and do not change as humans age (outside of damage to them)[0]. Humans only have 3 sets of these cones and do not gain or loose new types of cones as we age. The receptive bandwidths of the pigments are set by the biophysics of the molecules.
The perception of color does change as humans age, however. Typically, a newborn cannot yet use their eyes well. They can percieve light and dark, then red, then they can process depth, etc. This learning process takes a few weeks to months[1].
The cognition of color is a very in depth and facinating field. Thus far, it appears that color is a cultural and learned thing, in that if your language does not have a word for a color, it seems to be very difficult to determine the color as a different one from among other colors. Translating the mix of photons that your photreceptors are affected by into usable language is a very complicated process and is not well understood yet.
Fun Fact: The Illiad does not include the word blue and the Bible uses it very rarely as well.
[0] https://en.wikipedia.org/wiki/Cone_cell
[1] http://www.bausch.com/vision-and-age/infant-eyes/eye-develop...