To give no quarter means to show no mercy, to award no clemency. To give no quarter originally applied to military situations. If a victor was not willing to take care of prisoners he would warn that he would give no quarter.

What is the meaning of to give no quarter?

: no pity or mercy —used to say that an enemy, opponent, etc., is treated in a very harsh way The soldiers showed/gave no quarter to the enemy.

Where does give no quarter come from?

Etymology. The term may originate from an order by the commander of a victorious army that they will not quarter (house) captured enemy combatants. Therefore, none can be taken prisoner and all enemy combatants must be killed.