No. Peace Dollars ended in 1935 and Eisenhower dollars started in 1971.
1960 is not a rare date for Washington quarters. In average circulated condition, its value is determined mostly by the silver it contains. The price of silver changes every day, so check a newspaper or financial site for the current price, then multiply by 0.18 for a rough retail value. A dealer will pay half to 2/3 of that amount.
More:The value depends on where it was minted and what condition the coin is in. Also, this coin is composed of 90% silver so the minimum value is about $4 as of 01/2013. According to USA Coin Book, these are the values:
Philadelphia (no mint mark): In good condition, this coin is worth about $5.50 up to about $8.35 in MS60 mint condition. In MS65 brilliant uncirculated condition, this coin is worth about $23. Proof coins were also minted here and those are worth about $11.
Denver (small "D" mint mark): In good condition, this coin is worth about $5.50 up to about $8.35 in MS60 mint condition. In MS65 brilliant uncirculated condition, this coin is worth about $23.
Good question, but the penny that has ever brought the most money was the 1943 bronze, wheat penny. Fewer than ten of these were struck by error.
MoreIn 2011 a top-quality 1943-D bronze cent was quoted at $1.75 million. That's more than twice as much as the next-rarest US cents which are various varieties of Large Cents minted in 1793.
Please post a new question with the coin's date and whether there's a small S or D under the words FIVE CENTS on the back. If the date's worn off, it unfortunately has value only as a filler coin - maybe 10 cents or so.