From Latin 'cognoscere' (to come to know) — layered through Medieval Latin and Old French before reaching English.
To make someone aware of or familiar with something; to come to know personally.
From Old French acointier (to make known, to introduce, to bring into familiar knowledge), from Vulgar Latin *accognitāre (to make thoroughly known), from Latin ad- (to, toward) + cognitus (known, recognised), past participle of cognoscere (to get to know, to learn, to examine). Latin cognoscere is built from co- (together, thoroughly) + gnoscere / noscere (to come to know), from PIE *gneh3- (to know, to recognise). This PIE root is among the most productive in the family: it also gives Greek gignoskein (to know), Sanskrit
The English word 'quaint' is a sibling of 'acquaint.' Old French 'cointe' (known, clever, pretty) came from Latin 'cognitus' (known), the same root as 'acquaint.' Something 'quaint' was originally something well-known and cleverly made; the meaning drifted from 'knowledgeable/clever' to 'elegant' to 'old-