As noted by CoinTrackers.com, a 1962 U.S. Washington quarter is valued from $6 for a quarter in poor, or heavily worn, condition to about $17 for one in perfect condition. There were about 36,156,000 of these quarters minted.
This coin has a 90 percent silver content, which helps account for its relative high value. Quarters were made from a high silver content until 1965, when the rising cost of the metal led the U.S. Mint to make the coins from a more affordable combination of copper and zinc. According to an article from Wired, coin collectors especially prize pre-1965 currency, and many of these coins can still be found in circulation in 2014.