Colons frequently crop up as transitional punctuation preceding a quotation, but that particular punctuation mark is usually not a good choice, as explained in the discussions that follow the sentences below; a revision follows each discussion.
1. The graffiti included the words: “Black lives matter.”
This simple declarative statement requires no punctuation between the descriptive opening phrase and the quotation: “The graffiti included the words ‘Black lives matter.’” (There is an unlikely exception: The words have been previously alluded to, and now they are being explicitly stated. In that case, the colon is appropriate.)
2. Smith planned to head to the region immediately and promised people in the area: “No individual, no family, no community will be left behind.”
In journalism, a colon is often used to signal that a quotation is about to follow an attribution, but a comma is much more appropriate, because whereas colons generally punctuate with the halting force of a period, a comma is more smoothly transitional: “Smith planned to head to the region later Wednesday and promised people in the area, ‘No individual, no family, no community will be left behind.’” (If the attribution constitutes a complete thought, a colon is correct, as in this revision: “Smith planned to head to the region later Wednesday, and his promise to the people in the area was emphatic: ‘No individual, no family, no community will be left behind.’”)
3. The question is: “How did the outcome of World War I contribute to the advent of World War II?”
This sentence presumably refers to a written question on an examination of some kind, as in a high school history test, but whether it is a quotation or simply part of a narrative, a colon is obstructive (as explained in the previous item), and, just as a comma generally follows an attribution (such as “she said”) that introduces a quotation—again, see above—a comma is appropriate to separate the setup phrase here and the quotation: “The question is, ‘How did the outcome of World War I contribute to the advent of World War II?’” (If the question is merely posed in a narrative, rather than a reproduction of a written question, the quotation marks—and the capitalization of the first word—aren’t necessary: “The question is, how did the outcome of World War I contribute to the advent of World War II?”