"a member of the Christian denomination known as the Religious Society of Friends," 1651, said to have been applied to them in 1650 by Justice Bennett at Derby, from George Fox's admonition to his followers to "tremble at the Word of the Lord;" but the word was used earlier of foreign sects given to fits of shaking during religious fervor, and that is likely the source here. Either way, it never was an official name of the Religious Society of Friends.
The word in a literal sense of "one who or that which trembles" is attested from early 15c., an agent noun from quake (v.). The notion of "trembling" in religious awe is in Old English; quaking (n.) meaning "fear and reverence" especially in religion is attested from mid-14c.
There is not a word in the Scripture, to put David's condition into rime and meeter: sometimes he quaked and trembled, and lay roaring all the day long, that he watered his bed with his tears: and how can you sing these conditions (but dishonour the Lord) and say all your bones quake, your flesh trembled, and that you water your bed with your tears? when you live in pride and haughtiness, and pleasure, and wantonness .... ["A Brief Discovery of a threefold estate of Antichrist Now Extant in the world, etc.," an early Quaker work, London, 1653]
Figuratively, as an adjective, in reference to plain or drab colors (such as were worn by members of the sect) is by 1775. A Quaker gun (1809, American English), originally a log painted black and propped up to resemble the barrel of a cannon to deceive the enemy from a distance, is so called for the sect's noted pacifism. Philadelphia, Pennsylvania, has been known as the Quaker City at least since 1824. Related: Quakerish; Quakeress ("a female Quaker"); Quakerism; Quakerdom; Quakerly.