Diamond can say this based on two things. First, he can base the statement on the historical record. So far as we know, no hunter-gatherer society ever invented writing. Second, he has a theoretical reason why this would never happen. He says that writing is only needed by sedentary societies that produce their own food.
A relevant quote can be found on p. 236 in my book, about 2 pages from the end of the chapter:
Writing was never developed or even adopted by hunter-gatherer societies, because they lacked ... the institutional uses of early writing...
Why is writing necessary? It's first necessary for keeping records. That is how it always arose. Writing arose as a way of keeping records for the governments. These records tended to be records of how much of various supplies a government had and how much various people owned (so they could be taxed on it). A hunter-gatherer society has no need for such things. There are no large surpluses to be stored up in such a society. There is no government bureaucracy to keep track of such things. Therefore, there is no need to have writing.
It is also worth noting that Diamond says that the food surpluses are needed to feed the people who write. Writing was always the job of specialized people who had the time to learn to write. These people had to be fed from stored surplus food. They could not have the time to learn to write if they had to be out hunting or gathering. Therefore, food production was necessary in order for them to exist.
So, food production must precede writing for these reasons. The food is needed to feed the specialists who do the writing and there would be nothing to write about until food production created surpluses that needed to be recorded.