biggest medical employer here is Hamad Medical Corporation. Their website http://www.hmc.org.qa/hmcnewsite/ doesn't show any nurse positions under Vacancies, but I know that turn over is quite high so a CV sent to them will certainly get considered.
PS the reason turnover is high is to do with Asian staff using Qatar as a stepping stone to Canada, and/or getting married. I think morale and conditions are OK and I know that professionalism and training, sometimes lacking in the past, is being very strongly encouraged.