George Washington and Mozart lived in the same time period, but how far did Mozart's music spread in the 1780's-1790's? His music was popular in Europe at that time but would it have taken much longer to spread to America, or were there preformances of his music in America where it would have been possible for someone like George Washington to hear it?
Most likely. If he didn't hear it, at least it's pretty certain he had the option. Here's an article about music in Washington's time. Though the bulk of entertainment at the time was more opera and stage acting oriented, There were performances at the time of music by Handel, mentions of Mozart (mostly sonatas, etc) and other imported works.
A question was asked a while ago about what classical composers thought about the United States, that you may find somewhat relevant/interesting.