In episode 4 it was mentioned that Go Nam Mu became king in the year 179 AD.  Weren't he and Woo Hui already married at that point??? But in Woo Hui's Wikipedia page it says that those two married in 180 AD.  Can someone confirm which one's correct????? 

I advise you to not use Wikipedia as a source when it comes to historical figures and dates, especially for countries which are not the USA. It's often wrong or lack sources and explanations. ;)

King Gogukcheon was indeed crowned in 179.

'In the second year (cf. of his reign) (180), in the second month of spring, he established his wife, Lady U (于氏), as queen. The queen was the daughter of Je Na Bu (提那部) and Lady U So (于素)."

This is from the "Samguk sagi", the chronicle of the Three Kingdom, the primary source for events of that time. The date of the marriage was never mentioned, only that the king named his concubine queen in 180, so they were already married since before, but it's unknown when exactly.