NO (not in our universe, anyway)
double and triple bonds bond using pi bonds,
these sit above and below the axis of the sigma bond (sigma bonds are regular single covalent bonds)
for a double bond these pi bonds sit in above and below the axis
for a triple bond bond the pi bonds sit above and below AND in-front and behind the axis,
As we live in a 3-D universe (3 dimensions of space, ignoring time dimension), there are no other axes to use for any other pi bonds.
therefore a triple bond is the most you can get.
therefore no quadruple bonds. =(