Cities of Salt by Abdelrahman Munif is the closest the Gulf region has to a canonical novel — a five-volume sequence tracing the transformation of an unnamed Arabian country by oil, written by a Saudi-Syrian author who was stripped of his Saudi citizenship for the book's unflattering portrait of the oil era. Banned across most of the Gulf but widely read; the first volume is the most accessible entry point.
The Buried: An Archaeology of the Egyptian Revolution by Peter Hessler is not about Qatar but illuminates the cultural and political currents that shape the wider Arab world Qatar operates within. Hessler lived in Cairo for five years and writes about Egyptian society with the same close attention he brought to China.
Qatar: Small State, Big Politics by Mehran Kamrava (Cornell University Press) is the most rigorous analysis of how a country of 300,000 citizens parlays enormous gas wealth and a foreign policy of calculated ambiguity into outsized global influence. Academic but readable; the best single book on modern Qatar.