Is there a site in the network to ask about quotes?
Examples: "Who is the author of ...?" and "What's the meaning of ...?".
If not, is it worth proposing at Area 51?
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"What's the meaning of this quote" might be an appropriate question on ELL.SE. I know this since I'm a regular there.
Per our help center, questions that can be asked using a dictionary (called General Reference questions or GR questions) are off-topic and we have a close reason for them. So if you ask about the meaning of a quote, make sure your question can't be reasonably answered by consulting a dictionary. Please also take a look at our Details, please before posting.
"Who's the author of this quote?" though, is not a question that expertise in any field but Googling is required to answer. I don't recommend asking questions of this sort in any SE site, nor do I recommend creating an Area51 proposal for it.
Asking about the meaning of a given quote is on-topic English Language & Usage as well as on English Language Learners (both sites have tags for this: ELU, ELL). See this Meta question for deciding between the two. If the quote is in another language, the question is likely on-topic on the respective Stack Exchange, if one exist. Whereever you ask, please elaborate what you managed to understand by yourself and what exactly your problems are.
Asking about the origin of a given quote may be on-topic of the Stack Exchange for the language from which this quote originates. Here is an example for such a question on English Language & Usage. You should report your previous research, lest your question be closed. As such questions are rare, I do not expect any language site to have a policy regarding this (at least I could not find any Meta discussion on this on English Language & Usage or German Language). Thus, the on-topicness of such a question will probably be decided on a per-case basis.
If you want to understand the context of a quote, your question may be a good fit for History. They have a tag for quotes.
If you are looking for a specific quote but do not know its exact wording, it is almost certainly on-topic as a reference request on some related site.
If doesn't, is it worth to propose at Area51?
I think the existing sites cover this topic quite well and there are only very few people whom you could call experts on quotes. Thus, I do not think that a new site is needed for this.
Asking about the source of a quote is on topic on Literature Stack Exchange. See Is it on-topic to ask for help finding the source of a quote?. This category of questions is tagged with
quote-source, which currently has the following tag wiki excerpt:
Questions seeking to identify the source of a quote. If possible, include the exact quote whose origin you're seeking, or describe it as closely as you can. (For questions seeking to identify a entire story or work of literature from some remembered details, use the [identification-request] tag instead.)
Asking about the meaning of a quote is on topic on Literature Stack Exchange if the quote is from a work of literature. (The distinction between, on the one hand, questions asking to explain the meaning of a passage in a work of literature and, on the other hand, questions about the meaning of a quote from a literary work is not important.) See the tag wiki excerpt for
Questions regarding the meaning of certain terms or phrases used in a work of literature. If your question concerns the symbolic significance of something whose surface meaning is clear, use the [symbolism] tag instead. (...)
Since "literature" is impossible to define, you may get away with questions about the meaning of non-literary quotes, but that is not guaranteed. In addition, questions that can be answered by simple dictionary lookups may get migrated away or closed. However, the meaning of individual words and phrases in literary works often depends on context, and reading literature often means opening up meanings rather than finding "the one correct meaning" and this may be used as an argument against closing meaning questions. We have What do we think about basic [meaning] questions? on our meta site, but this is really an ongoing discussion.