It's impossible to say for sure, as there is no record of first use of this term. It predates the internet, though you find fewer uses of it before about the mid 1990s. It started as a public relations technique for working out in advance a list of questions likely to be asked by the media, and having possible answers ready.

It's really the internet that has made use of terms like Q & A or FAQ so common, as the amount of information people expect to find has grown so much through internet use. The sheer volume of people asking questions about any given topic has meant that all questions can't be answered individually and also that the same questions keep coming up over and over - so you create a list of Q & A or FAQ to deal with them. Until about the mid 90s it was common to hear sentences like 'There will be a question and answer session after the presentation' which would sound quite old fashioned now.