There are several gold quarters out there, but there is only one official gold quarter issued by the US mint. Any other gold quarter out there is gold plated.
The only official gold quarter in existence is the 2016 W Standing Liberty Centennial Gold Coin. The United States minted 91,752 of these coins to commemorate the 100th anniversary of the Standing Liberty quarter design. This coin is 0.25 troy ounces of 24 karat gold. Each coin is minted out of West Point and the W mint mark is located on the obverse side of the coin.
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Type: Standing Liberty Centennial Gold Coin
Face Value: $0.25
Composition: 99.99% gold
Total Weight: 0.25 ounces
This coin is worth at least its weight in gold. The gold melt value per coin is $447.91 as of October 23, 2021. This melt value is calculated from the current gold spot price of $1,791.63 per ounce.
The 2016 W Standing Liberty quarter is worth around $600 in SP 67 condition. The value is around $700 in SP 70 condition.
Gold-plated quarters are common. Over the years companies and individuals have taken standard quarters and coated them with a thin gold layer. In most cases the plating is 24k gold. It was common for companies to apply gold plating to state quarters when they were first issued in 1999.
In the United States it's not illegal to apply gold plating to a coin as long as the coin is not defaced.
Gold plated quarters are not valuable. The gold coated layer only contains a very small amount of gold. Therefore most gold plated quarters are only worth their face value of $0.25.