Quarterstone (クォーターストーン [quarterstone] in Japanese) is an area in western La Noscea in Final Fantasy XIV, facing the Rhotano Sea to the south. Skull Valley is to the north-west, and to the east is Three-malm Bend.
Quarterstone was the site of a moderately flourishing farming community before the Calamity struck, despite its soil consisting of too much sand and rock for most varieties of wheat to survive in. Special varieties of wheat were cultivated by the families to counteract this, and the common livestock in the area was dodos. However, the farmer families fled to Limsa Lominsa during the Calamity1 , leaving their fields and livestock to fend for themselves. It is only recently that they have started returning.
Note that while this area has been retconned in starting with version 2.0, in version 1.x it was still just another stretch of the greater Skull Valley area.
The Brewer's Beacon (ブルワーズ灯台 [brewer's toudai] or 'brewer's lighthouse' in Japanese) is a new lighthouse built immediately following the Calamity2 to replace Swiftperch Tower that was destroyed in the event. In addition to guiding ships safely to the nearby Aleport, it also serves as a watchtower against sahagin attacks and tidal waves caused by Leviathan.
The area known as the Flock (ドードー営巣地 [dodo eisouchi] or 'dodo nesting grounds' in Japanese) is what is left of the old fields that used to spread about Quarterstone. Feral dodos have taken the fields as their nesting grounds.
Note that Quarterstone was part of the Skull Valley area in version 1.x. Please see Skull Valley's page for the 1.x info.