Origins
Four is one of the oldest numbers with a name, and quarter is its direct descendant. The word comes from Latin quartus ('fourth'), from PIE *kʷetwer- ('four'). Old French quartier meant both 'a fourth part' and 'a city district' — because medieval towns were typically divided into four administrative zones.
The word's versatility is extraordinary. A quarter of a year is three months (the financial world runs on quarters). A quarter of a city is a neighbourhood (the Latin Quarter in Paris, the French Quarter in New Orleans). Military quarters are lodgings — from the practice of dividing a conquered town into four sectors and billeting troops in each.
The phrase 'give no quarter' comes from ransom practices. Captured soldiers could be spared by paying a quarter of their annual pay as ransom. 'No quarter' meant no ransom would be accepted — you would be killed regardless. Mercy and fractions share the same word.
Latin Roots
The most surprising relative is quarantine. Italian quarantina meant 'a period of forty days', from quaranta ('forty'). During plague outbreaks, Venice forced arriving ships to wait forty days before docking. The quarant- stem descends from Latin quadrāgintā ('forty'), built on the same root as quartus.
Four also gives us quart (a quarter of a gallon), quartet (a group of four), and squad — from Italian squadra, originally a 'square' formation of soldiers, from the four-sided shape.