That POA will be dead soon so dont do anything right now. DaveK will be on shortly to give you some zoysia info. But I will tell you that your lawn is just now coming out of dormancy and once it starts growing it should choke out any weeds you may have. An application of Pre-M will prevent that POA from germinating next winter so you may consider that.
That POA is not that bad, I have seen much worse.
Georgia Applicator License Number 09102
"A well-regulated militia being necessary to the security of a free state, the right of the people to keep and bear arms shall not be infringed."