No. Carbon does not exist as C₂. It naturally occurs as C. C has 4 electrons in its outermost shell (electronic configuration of C = 2, 4) which can be shared covalently with four other atoms to have a stable octet. On the other hand, nitrogen exists naturally as N₂. Each N is bonded to the other N by a covalent linkage. Apart from this linkage, each N can form 2 more bonds, that is overall each N can have bonds with three other atoms. This is because the last shell of N has 5 electrons, and it needs three more electrons to form a stable octet (electronic configuration of C = 2, 5).