Dear
I am too working in the Government Sector here in Qatar (MOI) so I am confident enough that NOC is a dream which will never come true.
Secondly if there is no job satisfaction then you should move on but not within Qatar. Instead I suggest you try in other GCC countries like UAE but make sure you clear all formalities in Qatar for exit. Please don't leave behind any pending loans, post paid services, etc.
NOC from Qatar is not a concern if you want to work in other GCC countries.