Actually, most people didn’t.  Indeed, most people in the ancient world could not write at all.  More than 90% of the population was illiterate.  And among those who could write, keeping a chronicle of their activities was not at all common; in fact it was almost unheard of.  First of all, paper (papyrus or parchment) was both expensive and difficult to obtain.  Secondly, there is simply no evidence of the popularity of the practice Brown describes.  If his claim was true, we would expect that in the surviving documents from the ancient world, there would be many examples of the kinds of “chronicles” he cites.  There aren’t.

It surprises some people that Jesus did not apparently write anything himself.  But it doesn’t surprise scholars of religion.  Many of the most influential people in antiquity did not write a word, but spread their ideas and teachings orally.  These teachings and ideas were only set down in writing by their followers, sometimes generations later.  Included on this list would be figures like Socrates, the Buddha, Confucius, and Muhammad.  Why should it be so surprising that Jesus was any different? 

Finally, there are no reputable “Q” scholars who claim that this hypothetical gospel was written by Christ himself.  There’s just no evidence to support such a claim.