The source of a quote that was written (perhaps in a book) would be the "author" of the quote. What if the quote was said? Is it the "dictator" of the quote? That sounds odd and "speaker" just doesn't sit right with me. Any other suggestions?
The long and short of it:
Far more commonly, and narrowly, defined as
1 A writer of a book, article, or document (ODO)
Nearly all definitions in most of the major dictionaries include "writing" as an essential part of the definition. Secondary meanings given are generally in the nature of metaphorical or overly-literary/ formal use that is an overly broad
1.3 An originator of a plan or idea (ibid)
that is not necessarily even about a literary piece, let alone a quotable one.
Wikipedia, on the other hand, places the broader definition first:
An author is broadly defined as "the person who originated or gave existence to anything" and whose authorship determines responsibility for what was created. Narrowly defined, an author is the originator of any written work and can also be described as a writer.
But again, it is not necessarily about a literary work in any way.
In summary, the word author cannot mean "the originator of an utterance," although it does not exclude such an instance.
We may thus have to look at alternatives such as source and speaker as a compromise. (Both seem inadequate to me, but source is widely used.)
Finally, a last resort could probably be:
New sets of words, new collocations of old phrases are new, whatever the expressed intention of the utterer. (OED)
As @Armen wrote, author is appropriate since it does not necessarily refer to writing — although not all dictionaries list this (Oxford doesn't, Cambridge has 'a person who begins or creates something'). Indeed, the etymology is (according to etymonline.com) :
author (n.) c.1300, autor "father," from Old French auctor, acteor "author, originator, creator, instigator (12c., Modern French auteur), from Latin auctorem (nominative auctor) "enlarger, founder, master, leader," literally "one who causes to grow," agent noun from auctus, past participle of augere "to increase" (see augment). Meaning "one who sets forth written statements" is from late 14c. The -t- changed to -th- 16c. on mistaken assumption of Greek origin.