Um, well… The short answer is…No. The quarters and dimes we use contain NO silver. None of the currently-minted quarters and dimes in circulation do! I was shocked, SHOCKED, I tell ya!
The slightly longer answer is that modern “clad” coins are minted by layering a mix of 75% copper and 25% nickel (aka cupronickel) over a pure copper core. For those that care about such thing, the final composition for a clad quarter is 91.67% copper and 8.33% nickel.
No silver, whatsoever.