how can a monarchy country become democratic
Answers
In common usage, a republic is defined as a country without a monarchy, so no.
However, if we go back to the original meaning of republic (res publica; a public thing), it could be used to refer to any country in which the government is based on public laws, rather than personal allegiance. In this sense, countries such as Denmark or the Netherlands would count as republics, because their monarchies and the rights and responsibilities thereof are clearly established in the constitution, but Saudi Arabia would not because every single feature of the Saudi state is defined with respect to the Al Saud family.
North Korea and Nazi Germany are mixed cases because, although republican in origin, they soon came to demand absolute personal loyalty and obedience towards a single individual. Britain is a mixed case of a different kind because, although everything is done in the name of the monarch and oaths of allegiance are made to her, she is widely recognised as a figurehead and independent from the real business of government. Anyway, it is her office rather than her person that matters.
Essentially, there is a broad spectrum between a pure direct democratic republic at one end and a feudal monarchy at the other. Every single country today exists somewhere in between.