quarter round comes in several sizes. it is cheap and usually always replaced on floor jobs because many floor contractors remove it so the flooring can go up and under the quarter round to create a nice finished look.
The quarter round should, I repeat should, lie flat and on top of the new flooring. This will make for a nicer look and also keep debris from entering the crack.
I am amazed that the floor contractor did not finish his job in installing NEW quarter round. However, you can do this IF you are knowledgable in cutting inside and outside 45 degree angles. If you are not, DO NOT ATTEMPT this.