A quaking bog is a bog in which the peat is saturated with water (or may even be formed over the surface of a shallow layer of water), and so it quakes and shakes when you walk on it. You may be able to make nearby trees rock and sway when you walk.
A closely related term is the German Schwingmoor, meaning a floating bog. It is entering the English language to distinguish saturated bogs from bogs that are actually floating on a layer of water. Schwingmoors are covered by a floating mass of peat that covers the body of water, and they often have many species of plants and moss growing on them.
Bogs are acidic; fens are alkaline (this may not always hold true in common usage). You might occasionally hear the term quaking fen, but even scientists will usually call a 'quaking fen' a quaking bog. That which quakes is a bog, even if it is alkaline. What a crazy world we live in.
Quaking bogs can be found in America, England and Europe, Antarctica (where it's called 'featherbed', and is not a proper bog), and South America (el tremedal in Spanish). If they exist in Australia, Asia, or Africa I have not been able to find reference to them; but I would be surprised if they did not.