What is the correct way to write birthday greetings “Happy birth day” or we can write “Wish you a Happy Birth Anniversary” to living person?
I'm never clear what people mean by "correct", but the word is "birthday" (one word).
I have never heard "birth anniversary"; but looking on GloWbE, I see that it does occur 495 times (as aginst 62005 for "birthday"). Looking further at those 495 instances, I see that 202 are from Indian sites, 94 from Bangla Desh, and nearly all the rest are from Sri Lanka, the Philippines and Pakistan. So it appears that "birth anniversary" has some currency in Indian English (understood broadly) and perhaps Philippine English, (is that a thing?) but is almost unknown in other English speaking areas.
In Britain, the conventional greetings are "Happy birthday", and "Many happy returns of the day" - I think the second one is perhaps a bit old fashioned now.