Can carbon have quadruple bonds?
s orbitals have spherical symmetry, and p orbitals have a dumb-bell shape.  When these orbitals overlap, then they form the bonding orbitals (and corresponding antibonding orbitals).  The p-orbitals are aligned along Cartesian (xyz) axes.  If all three p-orbitals overlap, then the s-orbitals can't get 'close enough' to overlap and form a fourth bond.  If we start mixing s and p orbitals, the geometry always puts electron density away from the other bonding atom.

Concisely: there are no arrangements which allow overlap of atomic orbitals that create four bonding molecular orbitals with the proper orientation.

C2 as a molecule can exist, it will simply only have 3 bonds plus two unpaired electrons confined to each carbon.

If you want a more elaborate answer we have to unpack the heavy quantum mechanics, and you won't be seeing that until your junior year.