No....but if massive enough when they collapse they could very well be a black hole one day. Our sun will go Red Giant one day but it is not massive enough to be a black hole...it will become a white dwarf about the size of Earth.
A lot of answers on this one, a "red giant" is a star at a particular stage of it's life. It is burning more than just hydrogen for it's fuel, that's why stars go through stages, they burn heavier and heavier elements and appear differently to us. Eventually, there will be nothing left to burn, it will burn out and become a "white dwarf". There may be a certain stage in a star's where it is so huge (!?!?!!!) that it pulls in all matter, even light, and becomes a "black hole". Scientist's haven't yet published much work on black hole's life, that's about all we know, other than there may be big and little ones.