not in quaternary, the making that is. The easiest gate to make is OR, but NAND is pretty simple too. The real trouble with XOR is, how exactly will it work? The AND, OR and NOT are pretty simple, but XOR in base 4 gets a little tricky.
AND - Lesser input ex 2 and 1 = 1
OR - Greater input ex 1 or 3 = 3
NOT - Inverse of input ex not 0 = 3, not 1 = 2
The real trouble comes with the expansion to a higher range of numbers.
XOR isn't exactly what it looks like. It's actually true when the number of true inputs is odd and false when the number of true inputs in even. Try using that