In a food processor or blender combine half of the cubed watermelon, half of the quartered strawberries, and half of the agave nectar. Cover and process or blend until smooth. Transfer to a pitcher. Repeat with the remaining cubed watermelon, quartered strawberries, and agave nectar.
Add lemon juice to mixture in pitcher. Chill for 4 to 24 hours or until very cold.
To serve, in a very large pitcher or punch bowl combine the fruit juice mixture and the chilled club soda. Serve over ice. If desired, garnish with watermelon wedges and whole strawberries.