Ken Larsen's web site - Zoysia lawn
Zoysia is a summer grass. It's green from mid-April to mid-December. Then, it's brown/dormant from mid-December to mid-April. It's best planted in early May. It needs a lot of sunlight and does not grow in shade.
Zoysia does not have a seed. You have to buy it as sod and transplant it into your yard. Super-Sod of Cary sells zoysia sod. [details] Super-Sod sells a pallet (500 square feet) for $ 360 as of February, 2020.
I bought my sod from a farm east of Fayetteville in the early 1990s.
Once planted, zoysia will slowly spread. In doing so, it crowds out weeds.
I fertilize it a few times/year. In March and April I will pull up weeds that pop up in those months ... before the zoysia rebounds from its dormant period.