No, you don't need to put "city", only Beijing. We only add city where it is actually part of the name, and that's rare (usually it is where the city shares a name with the country or state, for example New York City, Panama City, Mexico City). In English, Beijing is just Beijing.
oh... I often see "Beijing" alone without "city," but I found other cities, for example, Shenzhen, is sometimes followed by the word "city." I am not sure if it has something to do with Chinese city naming system.
Typically it would be unnecessary to add "province", unless it helps to resolve an ambiguity with "city".
We also sometimes (probably more often) say "province of X" or "city of X" instead of "X province" or "X city".
In some languages a generic label is required for use with place names. English is not one of them.
One exception that you might have encountered in 'London town', sometimes used in songs or in a whimsical fashion. This is the only exception that I can think of if we discount instances where you need to distinguish between the city and the county, province etc.
Irish counties are also a bit of a quirk in that they are often prefaced by the generic: County Clare, County Tyrone, County Wexford etc.
We would never say Nebraska state or Queensland state or Kent county (in the UK*), but as we move into less familiar countries or names, the qualifier is sometimes used: Madhya Pradesh state or Guandong province. Even Manitoba province, which shouldn't count as less familiar.
* US counties are actually called things like Kent County.