Quash means to put down, stop, extinguish, and it’s usually used to talk about ideas, feelings, or political movements. You wouldn’t quash a grape underfoot; you would squash it. But if you were a military dictator, you would quash a revolution.
Quash is an extreme word. It comes from the French word for smash, or shatter. If something is quashed it is completely suppressed, usually by something or someone very powerful or authoritative. If you wrote a poem and asked your favorite teacher to read it, and that teacher tore it to pieces, then your hopes were most likely quashed.