There is no government or official cellphone directory. There are a number of websites that offer search options by name or number, but the accuracy of these websites is mixed because they are based on public records.
Cellphone carriers do not publish listings of this information because it is considered private and is not acceptable for them to share without users' consent. Directory websites based on public documentation have spotty success records because some people do not make such information public and thus are not likely to be listed. When using such a site, the option to search by number is often more accurate than a name search.