What to do?
Full implementation of a mature GPIS / GPM system is a big job. The natural solution is to first implement partial improvements, and after so to focus on full improvement of these systems. You can start your work from the selected area of the entire system. Maybe you feel that you need to define project governance processes in your country? Good area to start with. Or you think it is necessary to establish a whole-government project management office? Or maybe you would like to start with training people who in the future would be the leading cadre of managing government projects? Another possible beginning of working towards GPM3 may be implementing the best practices in one selected sector of government's operation. Railways? Health care? Education? This would be a step towards the future achievement of the GPM3 Governmental level. You can start this way, but – if you are thinking about the overall improvement of GPIS – then you should plan from the outset what will be the final shape of this system supporting the work of your government.
Surely you should realize that "quick" is relative to the duration of the whole GPM3 implementation. This will take probably much more time than weeks.