It is not possible for anyone to know who discovered carbon. Carbon has been known from pre-historic times when it was used as charcoal to create fires. The name carbon for the element is derived from "carbo" which is what burnt wood is called in Latin.
Charcoal is one of the forms in which carbon occurs naturally, some of the others being diamond and graphite.
Carbon was first recognized as an element by Robert Boyle who noted that it was a substance that could not be decomposed into simpler substances.
Graphite and diamond were not considered to be forms of carbon for a long time. Antoine Lavoisier showed that diamond was a form of carbon in 1772 and Carl Wilhelm Scheele showed that graphite was also a form of carbon in 1779.