A great deal depends on the size and type of potato. One big russet can fill a quart. Two all purpose potatoes or a handful of fingerlings. Also, if it's a quart of whole, sliced, cubed, diced, mashed. Each will provide a different answer.
Not a very good recipe to give such a vague quantity.
According to some colonial/old time recipes I looked up, the potatoes, when measured by quarts, are rough sliced, or cubed (1-1/4" to 1-1/2"), or mashed (I guess because they take on an extra volume of air).
In most cases, they correspond directly to an equal volume of potato salad, which seems to be some sort of standard measure.
In any case, in all of the researching I did in books and on-line, the measure is of the final processed product - anything except whole potatoes.
This must be referring to a canning recipe. One I have for potatoes says "2 1/2 lbs potatoes per quart jar (to 3 lbs)" which is an estimation only of the amount of potatoes you should plan on having available per quart jar being filled. There is no way to be accurate since all potatoes are slightly different in size. In recipes like this you just keeping filling jars until you run out of potatoes.