In fact, formally, "to quote" is a verb, and "quotation" is a noun. And so it was until the end of the 19th century - the word "quote" was used exclusively as a verb. However, in 1888 the first case of using it as a short for "quotation" was registered. This transformation of the word initially met with strong disapproval in certain circles. However, it was so successful that over time it became widespread and began to be used more and more often.
To date, in an official and / or academic context, this reduction is still unacceptable. However, in informal speech and writing very often the word "quote" is used precisely as a noun - a short form of the word "quotation" or "quotation mark" (quotation mark).