The use of the words quote and quotation interchangeably has become prevalent in modern writing and has been allowed by most dictionaries although some may consider using quote in place of quotation as informal. However, in their traditional uses, these terms are slightly different.
The term quote as a noun traditionally refers to “a quotation giving the estimated cost for a particular job or service.”
Parents chip in to rebuild school playground at under 2% of quote
Stuff.co.nzDoctors quote big for rural posts
The HinduJacksonville lawyer sounds alarm on hefty quote for public records
News4Jax
However, it is now used as a noun meaning “a quotation from a text or speech.”
New Year 2017: Top quotes and messages to celebrate the special day with friends and family
International Business Times UKMiley Cyrus Explains Her Sobriety With a Justin Bieber Quote
TIMESome people are angry about the Jane Austen quote used on the new £10 note
Business Insider
Quote can also be used as a verb to mean “to repeat or copy out a group of words from a text or speech, typically with an indication that one is not the original author or speaker.”
RLA work quoted in Commons Library Paper
RLAPapers quoted by Rahul Gandhi name not only Modi, but two CMs and one Delhi ex-CM
Economic TimesTheresa May quotes murdered Jo Cox in New Year message calling for Brexit unity
Evening Standard
On the other hand, the word quotation is generally used as a noun referring to “a group of words taken from a text or speech and repeated by someone other than the original author or speaker.”
Jane Austen banknote unveiled – with strange choice of quotation
The GuardianA Pseudo-Darwin Quotation
National Center for Science EducationQuotation of the Day: There Was, Literally, No Place Like Her Self-Cleaning Home
NY Times
In summary, you can use both quote and quotation interchangeably to refer to words taken from a text or speech. However, you may opt to choose quotation in formal writing as some editors are still very particular with the traditional sense of these words.