First things first--a bit of research has turned up some stuff I should have mentioned earlier, but so it goes. Greek Κυδωνι (kythoni) should hardly be a surprise. Catalan codony/er had me confused until I found Friulian Italian mêl codogn for the fruit and codognâr for the tree. Hopes of further Romance languages that kept the dental stop were dashed by Sardinian mela pirongia "pear apple." And I can't even find anything for Occitan to see if there's a continuum from northern Italy to Northern Spain. Rounding out this odd mix is Lithuanian paprastoji cidonija, "Cydonian...uh...pear something?"
On to Slavic. My first Slavic love, Czech, gives us kdoule. I'd be able to tell you more about that if I hadn't mailed my Czech etymological dictionary off to Oregon or Vermont or somewhere. And that's the only etymological dictionary I actually own. It's clear that you lose that first vowel, though. In a situation we only see elsewhere in Romanian, that /kd-/ is pronounced /gd-/, through regressive assimilation of voicing. Czech has a non-productive suffix -oň giving us names of trees, and this leaves us with kdouloň, which I find to be a particularly beautiful word, although I bet that n won't format properly.
Slovak preserves the initial vowel in kutina. That's about as easy as it gets with Slavic, unfortunately. I don't know what to make of Polish pigwa at all. Serbian/Bosnian/Croatian have (normalized to Roman script) dunja, which is possibly a continuation of cydonium with the first syllable dropped altogether. (That is, like Czech, but with the difficult cluster simplified). Or it could be something else altogether.
Russian and Ukranian, thanks to contact (read: subjugation) by the Turkic tribes to the south and east (incidentally the neighborhood where the quince originally came from) have айва (aiva)--cf. Turkish ayva, Azerbaijani heyva. Thanks, Nate!
These discontinuities all make a certain amount of sense--there was a proto-Romance (i.e., Vulgar Latin) word for quince, and most of the remaining languages show reflexes of this, with some room for phonetic and semantic variation. The Germanic languages were more closely neighboring the classical languages of antiquity and in some cases (Charlemagne discusses quinces in his treatises on gardening) assuming their role and borrowed the term more or less uniformly. Where the Slavs settled in areas associated with classical languages and quince production (Czech/Slovak, the Balkan languages) they picked up some variant of it (if my supposition for the Balkan languages); Polish is a wildcard and the East Slavic languages, in spreading to the South, got their term from the Turkish.
I'm finishing this batch of membrillo, or quince jelly, or quince paste, or quiddany, or cotignac, or what have you, today. Quince on crackers with cheese, as it should be. Enjoy. From Greek to Latin to Old French to English to me to you.