What is the country with an area code of 823?

There is no area code 823 in North America (country code +1 = USA, Canada, etc.).

Country code +82 is South Korea.

Either you received a call from South Korea, or the caller ID was incorrect, either because valid data got mangled on its way to you or because a telemarketer or scammer sent fake caller ID data.

(The plus sign means "insert your international access prefix here." From a GSM mobile phone, you can enter the number in full international format, starting with the plus sign. The most common prefix is 00, but North America (USA, Canada, etc.) uses 011, Japan uses 010, Australia uses 0011, and many other countries use different prefixes.)