The two words quantitative and qualitative refer to two very different approaches to research. Qualitative research, to put it simply, relies on the thoughts, words and feelings of people to capture bigger ideas and concepts. Quantitative research, on the other hand, is an approach to research that relies on figures, numbers and statistics.
Clearly, given the fact that Diamond is looking into the ancient past to try and work out the response to Yali's question, he is unable to go back and interview participants. Instead, as the frequent charts, graphs and tables show, he relies on quantitative evidence to lead him to the various conclusions that he makes in his book. For example, you might consider the table that lists the different species of animals that can be domesticated and where those animals are. In the same way, have a look at the table that lists the plants that can be used to produce food and their location. This is the kind of quantitative evidence that Diamond welds together to answer Yali's question in this excellent book.