1960 was one of the final years that US silver quarters were minted. These coins 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 1960 quarter with no mint mark and also the 1960 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 scrap silver value for this coin is $3.22 as of September 19, 2019. This scrap value is calculated from the current silver spot price of $17.83 per ounce.
The 1960 quarter with no mint mark has a small amount of numismatic value. It has the lowest mintage of any silver quarter minted in the 1960s. In grades up to extremely fine condition, it will only sell for a slight premium above the silver scrap value. But in uncirculated condition, these coins each sell for around $7-10.
The 1960 D quarter is less rare and thus not as valuable, even in uncirculated condition. This coin in uncirculated condition only sells for a $1-2 premium above the silver scrap value.
Proof coins are available and they are valued at around $15 each. Graded proof coins with a grade of PR 67+ can sell for more. There were 1,600,000 proof coins minted so they aren't rare.