Yes, they were silver before the Coinage Act of 1965. So all dimes, quarters, half dollars and dollar coins were silver before 1965. The Kennedy half dollars were still 40 percent silver until 1971, and some of the Eisenhower dollars before 1976 also had 40 percent silver content. Now, the only silver coins the US mint makes are the ‘silver proof’ dimes, quarters and half dollars that you can get in proof sets. In addition, during world war II the Jefferson nickel had 40 percent silver content (otherwise the nickel has never had precious metals in it).
All 1965 dated quarters, such as the one you found, all have copper within rather than silver. As for finding silver coin in circulation still, that is highly unlikely. I last found a silver quarter (dated 1963) in change back in 1989.