I’ve seen several places in the note taking or zettelkasten communities the general advice that one should not include quotes in or amongst their notes. The general source of this “rule” seems to stem from Sonke Ahrens’ book Smart Notes. However, suggesting that Ahrens has a “rule” against quotes is a dramatic misreading of his intent. I imagine that this potentially stems from someone reading and excerpting his intent incorrectly and then others passing it along indiscriminately in the dreadful litany of one-page blogposts about how one should keep a zettelkasten.
The word “quote” (or close variations like “quotes” and “quotation(s)”) only appear 19 times in the first edition of Ahrens’ book.
In most of the contexts which have what one might call an “anti-quote” connotation, he’s directly recommending against the practice of indiscriminate highlighting/excerpting and collecting of general quotes specifically because they don’t aid in creating understanding by the reader. Instead he repeatedly recommends that one internalize the information by rewriting it in their own words instead. This helps the reader to better understand and know what the author is trying to convey. This also allows the reader to have material in their collection already written in their own words for later reuse.
Talking about “literature notes” Ahrens writes:

He did not just copy ideas or quotes from the texts he [Luhmann] read, but made a transition from one context to another.

Be extra selective with quotes – don’t copy them to skip the step of really understanding what they mean. Keep these notes together with the bibliographic details in one place – your reference system.

Places where quote appears in a context which argues against indiscriminate collection of quotes:

A typical mistake is made by many diligent students who are adhering to the advice to keep a scientific journal. A friend of mine does not let any idea, interesting finding or quote he stumbles upon dwindle away and writes everything down.

As well, the mere copying of quotes almost always changes their meaning by stripping them out of context, even though the words aren’t changed. This is a common beginner mistake, which can only lead to a patchwork of ideas, but never a coherent thought.

It is so much easier to develop an interesting text from a lively discussion with a lot of pros and cons than from a collection of one-sided notes and seemingly fitting quotes

Even doctoral students sometimes just collect de-contextualised quotes from a text – probably the worst possible approach to research imaginable.

It is not surprising, therefore, that Lonka recommends what Luhmann recommends: Writing brief accounts on the main ideas of a text instead of collecting quotes.

Now let’s take a quick look at some of Ahrens’ “pro quote” passages which provide the opposite view of when and where quotes can be useful:

The available books fall roughly into two categories. The first teaches the formal requirements: style, structure or how to quote correctly.

It would certainly make things a lot easier if you already had everything you need right in front of you: The ideas, the arguments, the quotes, long developed passages, complete with bibliography and references.

You follow up on a footnote, go back to research and might add a fitting quote to one of your papers in the making.

In this textual infrastructure, this so-often taught workflow, it indeed does not make much sense to rewrite these notes and put them into a box, only to take them out again later when a certain quote or reference is needed during writing and thinking.

How is one to have useful/impactful/fitting/necessary quotes at hand if they haven’t excerpted them as they read? In these portions he is actively suggesting that quotes from one’s reading in their notes can be a good thing and can help in making persuasive arguments. The secret is that they need to be done judiciously. One needs to be able to quote in a manner which keeps the original context and argument, but which can also fit into your current context and provide support or further argumentation.
As an example of terrible decontextualization, who hasn’t attended a wedding that featured a reading of 1 Corinthians 13? The passage seems wholly appropriate for a church wedding reading, but when you consider that it’s excerpted out of context you might reconsider using it at your own wedding. Go back and try reading it in light of being sandwiched between Corinthians 12 and 14 and you’ll change your mind that chapter 13 is about the sort of romantic love and implied by a wedding. Once you’ve done this, there’s added comedic subtext to scenes like the following from Wedding Crashers (New Line Cinema, 2005):

Father O’Neil: And now for our second reading I’d like to ask the bride’s sister Gloria up to the lectern.
John Beckwith: 20 bucks First Corinthians.
Jeremy Grey: Double or nothing Colossians 3:12.
Gloria Cleary: And now a reading from Paul’s first letter to the Corinthians.

To prevent embarrassment of this sort, perhaps when you’re quoting a source directly you ought to provide at least a short note about the context in which the words were provided?
Any good rhetorician will tell you that quoting works in your writing can be incredibly helpful in building context and creating authority.
If anything, Ahrens’ book is missing a section on “how to quote correctly”, and this is a stumbling block of his text. As a quick remedy, one could read a bit of Seneca perhaps?

“We should follow, men say, the example of the bees, who flit about and cull the flowers that are suitable for producing honey, and then arrange and assort in their cells all that they have brought in; these bees, as our Vergil says: ‘pack close the flowering honey And swell their cells with nectar sweet.’”
—Seneca in 84th letter to Luculius (“On Gathering Ideas”), Epistles 66-92. With an English translation by Richard G. Gummere. Cambridge, MA: Harvard University Press (Loeb Classical Library, 2006), 277-285.

Beyond just Ahrens there are several thousands of years of prior art seen in the commonplace book tradition where quotes feature not only prominently but at times almost exclusively. Quotes, particularly sententiae, are some of the most excerpted and transmitted bits of knowledge in the entire Western canon. Without quotes, the entire tradition of excerpting and note taking might not exist.
Of course properly quoting is a sub-art in and of itself within rhetoric and the ars excerpendi.
Fellow note taking writer Umberto Eco warns against this same sort of indiscriminate collecting without actively making the knowledge your own. In How to Write a Thesis (MIT Press, 2015, p125), instead of railing against indiscriminate highlighting, or digital cutting and pasting, Eco talks about another sort of technological collection tool more rampant in the 1990s and early 2000s which facilitated this sort of pattern: the photocopier.

Beware the “alibi of photocopies”! Photocopies are indispensable instruments. They allow you to keep with you a text you have already read in the library, and to take home a text you have not read yet. But a set of photocopies can become an alibi. A student makes hundreds of pages of photocopies and takes them home, and the manual labor he exercises in doing so gives him the impression that he possesses the work. Owning the photocopies exempts the student from actually reading them. This sort of vertigo of accumulation, a neocapitalism of information, happens to many. Defend yourself from this trap: as soon as you have the photocopy, read it and annotate it immediately. If you are not in a great hurry, do not photocopy something new before you own (that is, before you have read and annotated) the previous set of photocopies. There are many things that I do not know because I photocopied a text and then relaxed as if I had read it.

Many people may highlight, tag, or collect a variety of quotes within a text, but this activity is only a simulacrum of understanding and knowledge acquisition. This pattern can be particularly egregious in digital contexts where cutting and pasting has be come even easier and simpler than using a photocopier. Writing it down and summarizing important ideas in your own words will actively help you on your way to ownership of the material you’re consuming.
A zettelkasten with no quotes—by definition—shouldn’t carry the name. So let’s lay to rest that dreadful idea that quotes aren’t allowed in a zettelkasten.
And if you’re just starting out on your zettelkasten or commonplace book journey and don’t know where to begin, I’ve recommended before writing down the following apropos quote and continuing from there:

No piece of information is superior to any other. Power lies in having them all on file and then finding the connections. There are always connections; you have only to want to find them.
—Umberto Eco, Foucault’s Pendulum (Secker & Warburg)