I just encountered this construction on a UK site. Is it in common UK use? I never hear it here as a verbal form of "quiet." It sounds very Germanic to me.
Yes, that would be one use. Teachers often say to a noisy class "Quieten down now." (As cat says, the imperative) "Everything has quietened down now" would be another.
The verb is quieten and means to make or become quiet.
It can be used like any other verb although my dictionary does describe it as chiefly British.
"No man but a blockhead ever wrote except for money." Samuel Johnson.
Across the pond, our teachers would say, "Quiet down now!" The AHD says that "quiet" can be either an intransitive or a transitive verb. How then is 'quieten' different, or isn't it?
Very common once as a deadjectival causative: blacken, whiten, redden, gladden, madden, dampen, broaden, deepen, darken, lighten, brighten.
Also the odd denominal: lengthen, strengthen, heighten.
No longer a living suffix: we can't create new words biggen, smallen, bluen, etc. (pace the Simpsons). There is a nineteenth-century word 'louden', so that probably also doesn't occur in American dialects.
I think you have to make a distinction between the suffix -en that marks the infinitival verbal form in many Germanic languages (English lost its somewhere between Middle and Present-Day English), and the -en verbal inchoative suffix (lengthen, redden).
The Pennsylvania Dutch example is interesting because they've made a verb out of a preposition.