There are several synonyms for the phrase "keep quiet about something". One of the most commonly used is "keep something under wraps", which implies that the information in question is being kept secret from the public or certain individuals. Another phrase that can be used is "hush up", which suggests that the person speaking wants others to remain silent or stop talking about a topic altogether. "Keep mum" is another synonym that suggests keeping one's mouth shut and not revealing information that could potentially cause harm or embarrassment. "Maintain silence" is a more formal synonym that suggests a deliberate and intentional decision to remain quiet.