The answer is easy. Middle easterners want kids for cultural and religious reasons. I know any talk of religion and/or culture is verboten in modern American discourse but it's a significant factor in most the world. Yet it makes most Americans uncomfortable
It also helps that Qatar (along with the UAE and Kuwait) provide hefty benefits to married couples for marriage and childbirth.
Emirati and Qatari women have almost 50% more children than Saudi women despite cultural similarities because their government offers crazy good benefits.
Saudi Arabia has less money per citizen so families bear more of the cost and opt for smaller families.
Culture plays a huge role but so do government incentives.
Qatar is a pretty developed country, and has a higher than average female workforce participation. This is not to say it has perfect freedom for anyone. I honestly can't answer your question. These things remain to be seen.
