What is the date of the most valuable US quarter dollar?

That's an extremely broad question. Quarters have been issued since the 1790s so there are many valuable issues, and the "most valuable" title can shift as time passes. Some of the highest-value quarters are 1796, 1870-CC, and 1871-CC. You can find a full set of current prices at any number of web sites. One good portal site is http://www.coinclub.com/prices/