When watermelons are ready to harvest, vine tendrils will begin to turn brown and die off. If the tendrils are green, the melon is not ripe. A ripe watermelon will make a dull hollow sound when thumped. The soil side of a watermelon will turn from white to pale yellow when the fruit is ready for harvest.
For more information on growing Watermelon, check out this great guide from Harvest To Table: