WordNet records no synonym and no antonym for quaintness. It is the only member of each of its synsets, which is common for technical, taxonomic and single-sense vocabulary — there is no other English word that means the same thing. The broader and narrower terms below are the useful relations in that case.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
the quality of being quaint and old-fashioned
“she liked the old cottage; its quaintness was appealing”
strangeness as a consequence of being old fashioned
“some words in her dialect had a charming quaintness”
Synonyms, antonyms and the broader and narrower relations for quaintness come from Princeton University’s WordNet. Full sources and licences.