From ThoughtCo:
Often the words quote and quotation are used interchangeably. Quote is a verb and quotation is a noun. As A. A. Milne put it in a humorous note:
“A quotation is a handy thing to have about, saving one the trouble of thinking for oneself, always a laborious business.”According to the Oxford Dictionary, the word quotation is defined as, “A group of words taken from a text or speech and repeated by someone other than the original author or speaker.”
The word quote means to “repeat the exact words of another with the acknowledgment of the source.” In Ralph Waldo Emerson’s words,
Every book is a quotation; and every house is a quotation out of all forests, and mines, and stone quarries; and every man is a quotation from all his ancestors.
“Going Back to Roots: Origin of the Words “Quotation” and “Quote”
The origin of the word quote goes back to Medieval English, sometime around 1387. The word quote is a derivation of the Latin word quotare, which means “to mark a book with numbers of chapters for reference.”
According to Sol Steinmetz, author of the book, “Semantic Antics: How and Why Words Change Meaning,” 200 years or so later, the meaning of the word quotation was expanded to include the meaning, “to copy out or repeat a passage from a book or author.”
One of the most frequently quoted American personalities is Abraham Lincoln. His words have proved to be a source of inspiration and wisdom. In one of his many famous writings, he wrote,
“It is a pleasure to be able to quote lines to fit any occasion.”
Link to the rest at ThoughtCo
PG admits that he doesn’t like the word “quotation.” If he were grammar king for a day, he would permit “quote” to be used as a noun or a verb.
Part of this is more Mrs Grundyism, the “nominalizations are always bad” school of thought (to which my sarcastic but factual reply is “‘nominalization’ is a nominalization” — which, when offered to a legal writing instructor fresh out of a judicial clerkship who was well over a decade younger than I was and had no experience actually writing directives and instructions for “real people,” resulted in a full minute of silence). It’s also important to remember that there is not a complete congruence between “verb” and “predicate,” and that the identity and nature of what are usually called “objects” of the verb (not to mention The Evil That Is Passive Voice†) change both meaning and function. That nominalizations can and sometimes are used to deflect attention from responsibility does not make them universally misleading or otherwise improper.
† Mistakes were made in adopting this precept as a definitive rule of writing. For one thing, the “rule” presumes that the speaker correctly knows the full chain of causation and is in a position to so state.
Good news: the OP is full of crap. The claim it makes is true historically, but “quote” as a noun is fully standard, appearing in dictionaries without any cautionary text. Merriam Webster dates its use as a noun to 1888. This simply is something that peevers enjoy peeving about. They are not to be taken seriously.
A natural language is always moving along, and nothing about it is permanent. In the long run, only usage matters.
Unless, of course, your intended audience is the typical “my way or the highway” gatekeeper who has status issues.
Many people, myself included, are aware that “quotation” is a noun and “quote” is a verb, but I and many others use “quote” as a shortened version of “quotation,” as in “Quotes of the Day,” a regular feature of my instructive Daily Journal.