By some experimentation, we find the possible solutions:

\(\displaystyle (Q,D,N)=(1,1,5),\,(1,2,3),\,(1,3,1)\)

All 3 possible solutions have 1 quarter in the jar. Now we could simply reason as follows:

We know there must be at least 1 quarter in the jar, and there cannot be 3 or more since that's more than 60 cents. So, there must be 1 or 2. If there are 2, then that leaves 10 cents for the dimes and nickels, but with at least one of each, that's a minimum of 15 cents that must be left over. Thus, there can only be 1 quarter in the jar.