"Showing mercy" is the most appropriate antonym for the phrase "giving no quarter." It means to be compassionate, understanding, and forgiving towards others. Unlike giving no quarter, showing mercy is more about being kind and empathetic. Other antonyms include being helpful, accommodating, and conciliatory. All these words signify that the person is willing to compromise or negotiate, ensuring that both parties are satisfied. In contrast, giving no quarter means that the person is unwilling to compromise or show mercy. They do not offer any leniency or compassion, leading to a difficult or hostile situation. Therefore, using the opposite of "giving no quarter" can improve relationships and foster goodwill among people.