Zoysia is a warm weather grass that turns brown in cold weather. Looking on the net it is difficult to find a temperature range ; Apparently the sellers do not want to discourage anyone from buying it. So, likely it is normal and will green-up in warm weather, but the weeds will still be there unless you remove them.
When Zoysia is slow to get started for whatever reason then weeds will opportunistically try to get established. You may be able to examine each weed patch to determine if there is a central point from which each is growing, in which case remove the crown at soil level and the problem goes away for a while giving the grass time to re-establish itself.
We used a "manual weed cutter" when I was growing up. It's like a safer version of a scythe. I still use one to harvest my clover and Timothy grass for feeding to my rabbits. If you Google the quotes, you'll see a picture of it.