PVC Damage: 1932-D & 1932-S Quarter

Odd I thought even the oldest Whitman Folders were Blue.
Regarless, what you may have is from the glue used to make a Folder. The process is slapping glue on a sheet of paper and dropping the slotted holed cardboard on that. The rear of the slots is shinny from glue. Moisture activates this glue and it attacks coins. Many coins end up with all sorts of markings on the reverses from this.
Try a dip in Acetone. If it's from the glue, should just go away.

You wrote that you were inexperienced ...

so in case you didn't know, use only pure acetone. It is available at hardware stores.

In my opinion, you are conserving the coins and not cleaning them. Using silver polish to make them shiny would be considered damaging to the coins, as would a bunch of other things.

Experts can tell if a coin has been dipped (in acetone) but I don't think it's the end of the world for that coin if you get the glue off of it. If you ever decide to sell the coin, you could mention that you gave it a bath in acetone. The buyer would appreciate it because if the buyer finds out later they'll feel cheated.

Same here.

To answer the OP. I had a friends collection in the same type Whitman green folder. This was a complete set , early dates 30's through 50's /60's. I used two baby jars and let each coin soak overnight in pure acetone with the lids on then, pulled the coins out and redid in new acetone for 10-15 minutes. Let the coins sit on a paper towel overnight. Bought a new coin album and in they went.
Did this in a well ventilated garage and out of the sun.
Took awhile but no more PVC.
I'm not sensitive to acetone so no gloves. I did use a tweezers to move them.
Let us know your results.

Those are the before pics.

PVC isn't always green; heavy infestations (like seems to be described here) and early-stage infections are whitish/clear. And the use of acetone cannot be detected by any expert. The only possibility is if you use acetone to remove something which covered part of the surface of the coin, which then aged more slowly than the uncovered parts. The result would be pretty obvious, but no different than if you'd used water to remove it. That's why not even acetone is an "always" solution.

These coins leave no choice. Acetone, or worse happens. Even if it isn't PVC, the coins won't suffer for the try.

Emphasis is mine.

Possible (assuming one was somehow able to keep it from evaporating), but very, very improbable.

Agreed. It would have completely evaporated long, long before the coin made it to the TPG.