This was the second to last year that US silver quarters were minted. The 1963 quarters aren't rare, but they do have value as silver bullion.
Type: Washington Quarter
Face Value: $0.25
Composition: 90% silver, 10% copper
Silver Weight: 5.62 grams
Total Weight: 6.25 grams
Current Silver Bullion Value: $3.22
The US minted the 1963 quarter with no mint mark and also the 1963 D quarter. The mint mark can be found on the reverse side of the coin.
The value in circulated condition is worth at least its weight in silver. The silver melt value is $3.22 as of September 19, 2019. This melt value is calculated from the current silver spot price of $17.83 per ounce.
These quarters don't have much numismatic value because of how many were minted and how many are still around. But in uncirculated condition both the 1963 quarters with no mint mark and the 1963 D quarters each sell for a couple dollars above their scrap silver value.
Proof coins are available and they are valued at around $8 each. Graded proof coins with a grade of PR 67+ will sell for more. A little over three million collectible proof coins were minted which was a large number for the time period.