2½+1¾=4¼.
4¼=17/4. We can only use one 4 so only one of the numbers can contain an odd quarter, that is, ¼ or ¾. But if it contains ¼ then the other number can't be a whole number, so we need to choose ¾. The smallest mixed number is 1¾=7/4.
Now, subtract 1¾ from 4¼=(17-7)/4=10/4=5/2=2½. So we get 2½+1¾=4¼. The two numbers use the digits 1, 1, 2, 2, 3, 4, so this answer fits the requirements.
1½+2¾=4¼ is another solution.