No. Zika cases do NOT require quarantine, because it is typically spread from mosquito to person, not casual contact with other people. Quarantines are extreme measures used for serious or fatal diseases spread by direct human-to-human contact.
However, people infected with Zika must be careful to avoid mosquitoes. The most effective way to prevent future cases is to get rid of mosquitoes which might bite infected persons and then spread the virus to uninfected persons.