To start, I searched Google for a chunk of the quote. You have to be careful with such results because there are all sorts of "quote sites" out there that are unhelpful and even unreliable... but I saw a promising result from an old issue of the magazine Ebony. This is on the Google Books site, which has a number of digitized old magazines. The article tells us that the quote was originally published in an advice column that King wrote for a time in Ebony, apparently sometime in the mid- to late-1950s.
If all you needed was a reliable source for the quote, you could use this article. But why not find the original source that article mentions? I couldn't find it using Google Books, but I happen to know that we have Ebony in the library databases going back quite a while. So I went to OneSearch and used the Advanced Search page. In one box, I put the word ebony and set that to search Journal Title/Source. In the next box, I put a short part of the quote in quotation marks and set it to search All Text (meaning the full text of the article, not just the title, abstract and subjects):
This gave me just two results: the article we previously found using Google, and the original source of the quote, the advice column in Ebony from May 1958.
Now for your APA citation. OneSearch will generate citations, but sometimes they have problems. Here is what it gives you as APA:
KING, M. L. (1958). ADVICE FOR LIVING. Ebony, 13(7), 112.
The capitalization is clearly wrong, but also, since Ebony is a popular magazine, the month should be listed. And you are supposed to include a "Retrieved from" note for online sources. That becomes complicated because you are usually supposed to use the home page of the magazine or journal, but in this case, you can't get the article from the magazine's website. According to the APA Style Blog, in this case you actually use the name of the database and the accession number. So I think you'd probably do it this way:
King, M. L. (1958, May). "Advice for living." Ebony, 13(7), 112. Retrieved from MasterFILE Complete, 48347474.