Regardless of the time of day you decide to fly, the ticket price will remain around the same. Book flights from Qatar to U.S. Virgin Islands as you normally would.
This is defined as Doha Hamad Intl Airport (DOH) and Saint Thomas Island Cyril E King Airport (STT). Codes like Saint Thomas Island Cyril E King Airport are often used to just call airports uniquely. Ready for more IATA code knowledge? These are related airports for your current travel destination: Christiansted Henry Rohlsen Airport (STX), Saint Thomas Island SPB Airport (SPB), Christiansted Sea Plane Base Airport (SSB).