A standard 1965 quarter isn't a rare date. If it is overstruck with a nickel die, though, the value could be greater if it's a verifiable error. If the nickel image is reversed, it may or may not be an actual Mint error; if it isn't, it could've been done by anyone with a hammer after the fact, which would decrease the numismatic value (not to mention its ability to be used in a vending machine).
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It's copper-nickel not silver. 1964 was the last year for 90% silver quarters.
US quarters didn't contain any nickel until 1965. 1964 and earlier quarters were made of an alloy of 90% silver and 10% copper. Please see "What is the value of a 1941 US quarter?" for more information.
The 1965 Washington quarter was the first copper-nickel issue of this series and almost 2 billion were made, any coin found in circulation is face value. High grade uncirculated coins are only about $1.00 retail.
The melt value for a 1964 nickel is the same as any other date of nickels (except 1942-45), because unlike the dime and quarter, nickels weren't silver, and there was no change to it in 1965. As of 19 August 2013, U.S. nickels have a melt value of 4.6 cents.
around 50 cents