Bsrgvty

All languages have words that are unique and hard to translate, because most other languages don't have a perfect matching translation for it. The German word Gemütlichkeit is one of them. You can translate is as "cosiness, snugness, sociability, comfortability, warmth, friendliness, slowliness, homelikeliness, ..." but all of those words matches only parts of the meaning of Gemütlichkeit. Other German words that are hard to translate are fremdschämen, Schadenfreude, Nervensäge, Weltschmerz, Erklärungsnot, Mahlzeit!, Fernweh, Fiesling, Schönwetterfreund and some of them even are used a foreign words in English: doppelganger (from German Doppelgänger literally: double walker), kindergarten (Kindergarten = children's garden) and many others.

The adjective quirky is one of the English words that are hard to translate into German. Others are mind, awkward, doggy bag, no-brainer, serendipity, bromance.

All you can do with such words is to step away from the word-to-word translation level and enter the "meaning level". Some call it free translation. You have to analyze, what is the meaning and intention of the speaker or writer, and then you have to re-think this meaning in the other language and express this meaning and intention in the other language. If you have to express ideas, that are easy to express with one of the words discussed here in one language, you maybe have to build complete new sentences in the other language.

This is what makes translation so hard. Every language has a set of words, and this set limits the way of thinking. Other languages have different sets with different meanings. So, speakers of other languages have other limitation of how they think. (And multilingual people have less limits of thinking, because they have more possibilities to think.)

If you grow up in an environment, where being gemütlich (adjective) and Gemütlichkeit (noun) are part of your everyday life, and then you want to say: "Simon ist ein sehr gemütlicher Mensch" or "Dieses Café strahlt eine angenehme Gemütlichkeit aus." or "Ich trinke erst mal ganz gemütlich ein Bier." in an other language than German, then you will get into trouble, because the idea that is connected with Gemütlichkeit is spread over many different words in other languages, and you have to find not only the best matching word for the actual context, but you also sometimes must provide the right context, that already is built into the original word.

The concept of quirkiness does not exist in German. (It exists in variations, but not in exactly the same way as in English.) Every German translation that is sufficiently close to the original meaning has a negative connotation. To be quirky means to be out of the norm, and being out of the norm is negatively connoted in most situations.

I don't really know the English word quirky, so I have no idea how positive or negative connoted it is, but the phrase "I like her ..." (in German: "Ich mag ihr(e(n)) ...") already provides a very positive context.

So, although all nouns in this list are negative connoted in other contexts, the whole sentence still is positive:

Ich mag ihre Verschrobenheit.

Ich mag ihre Schrulligkeit.

Ich mag ihre Ticks.

Ich mag ihre Marotten.

Ich mag ihre exzentrische Art.

• 
  Verschrobenheit - eccentricity, crankiness


• 
  Schrulligkeit - crankiness


• 
  Ticks (plural of Tick) - quirks, spleens, tics, fads


• 
  Marotten (plural of Marotte) - foibles, whims, fads, quirks


• 
  exzentrische Art - eccentric nature, cranky nature, bizarre nature (also manner instead of nature)

I would use

Ich mag ihre Eigenheiten

it is relatively neutral in meaning so it doesn't have any negative connotation.
But of course it could be too neutral for "quirky".

Of course, Hubert Schölnast is right about the problems of translation in general, and he also suggests a few good attempts at translating quirky.

The first word that came to my mind though was ausgeflippt.

Ich mag ihre Ausgeflipptheit. Ich mag ihre ausgeflippte Art.

Used as a verb it can mean that someone lost their temper, but as an adjective it could also mean that a person is a bit out of the norm, unpredictable in a creative way, very active or of an agile mind, maybe a little hyped up. It's colloquial than what Hubert suggested.

Merriam-Webster's entry for quirky describes it as "unusual in especially an interesting or appealing way".

So one could simply say:

Ich mag ihre ungewöhnliche Art.

An even better fit, because it usually has a positive connotation even in German (I think Volker Landgraf has a point with Germans disrespecting deviations from the norm, except perhaps in Berlin) would be:

Ich mag ihre originelle Art.

In a humorous context I would use ulkig.

Ich mag ihre ulkige Art.

Some examples from Wiktionary:

Seine Kleidungsstücke passten farblich überhaupt nicht zusammen, das sah ulkig aus.

Renate ist schon eine ulkige Nudel, die gratuliert keinem zum Geburtstag.

and from Duden:

mit der Pappnase sah er wirklich ulkig aus

er ist ein ulkiger Mensch, Vogel

As a supplement of Hubert's answer, it is a social phenomenon in German(y) that beeing too different from others is generally regarded negatively, therefore you will hardly find any word for "quirk" that has a positve or neutral connotation unless combined with something like "I like...".

This is different from e.g. England, where it seems to be perfectly socially accepted that virtually everyone has a spleen or two.

Ich mag ihre unkonventionelle Art

... which puts an emphasis on defying convention without suggesting open rebellion. The extent of the defiance remains open to interpretation and in general the phrase suggests ingenuity, initiative, and thinking outside the box.

Credits

... go to @ChristianGeiselmann who noted the positive connotation of 'unkonventionell' in one of his comments.

It really depends on the context (as many others have pointed out). If you don't need to keep the same sentence structure, you could say "Ich mag, dass sie nicht 08/15 ist, das macht sie interessant!"

DeepL recommends "Ich mag ihre Eigenart." in the first try. If you are not satisfied with "Eigenart", it recommends "Macken" as the first alternative - which sounds better to my ears:

Ich mag ihre Macken.

If you don't like "mag", the first alternative is

Ich steh auf ihre Macken.

which sounds even better to me, but might express a little too much sympathy :) !

I would go with "eigenwillig" for "quirky" and "Eigenwilligkeit" for "quirkiness". Literally it means "with a mind of one's own". It doesn't reach into the negative as "Sturköpfigkeit", "boneheadedness" does and is more about a refusal to be pinned down into a category rather than refusing to consider changing categories.