the modern English word is a merger of two words, both in Middle English as quashen, from two unrelated Latin verbs.
1. "to suppress, overcome" (mid-13c.); "to make void, annul, nullify, veto" (mid-14c.), from Old French quasser, quassier, casser "to annul, declare void," and directly from Medieval Latin quassare, alteration of Late Latin cassare, from cassus "null, void, empty" (from extended form of PIE root *kes- "to cut"). The meaning "subdue, put down summarily" is from c. 1600.
2. "to break, crush, beat to pieces" early 14c., from Old French quasser, casser "to break, smash, destroy; maltreat, injure, harm, weaken," from Latin quassare "to shatter, shake or toss violently," frequentative of quatere (past participle quassus) "to shake," from PIE root *kwet- "to shake" (source also of Greek passein "to sprinkle," Lithuanian kutėti "to shake up," Old Saxon skuddian "to move violently," German schütteln "to shake," Old English scudan "to hasten").
In Medieval Latin, quassare often was used for cassare, and in later French the form of both words is casser. The words in English now are somewhat, or entirely, fused. Related: Quashed; quashing.