Must delurk for grammar-type things!
Much as it should appear the contrary, given that the word derives from the proper name Quixote (as amply mentioned above), the accepted pronunciation is, in fact, "quicks-otic." In case you don't believe dictionary.com or Merriam-Webster, the venerable Oxford English Dictionary includes only that pronunciation, as well (and they're the kings of including variant pronunciations with their entries). Since the OED is behind a paywall, I've attached a screenshot of their entry for "quixotic" for y'all's browsing pleasure
The best reason I can find for this pronunciation is that the adjective form was first used in English (in the 1700s), hence the more strident Anglicization. We mostly see these Anglicized pronunciations today with proper nouns, like Mexico and Paris (which few Americans would commonly pronounce as "Meh-hee-co" or "Pa-ree," unless adopting a deliberate affectation.)
Quixotic pronunciation discussion aside, the one that
really gets me is the word "forte," as in "strength or special skill." Most people pronounce it "for-tay," as one would for the dynamic notation in music (which is Italian!), but in fact, the proper American pronunciation is just "fort" (like what you'd build with Lincoln Logs, no second syllable accent). Since SO many people have continued to use the "for-tay" form, this pronunciation is beginning to be included as a variant in most dictionaries. Granted, "figuratively" has been added as a variant definition for "literally" in some dictionaries in recent years, so I don't even know if I can trust dictionaries any longer! (Also, if you want to be a SUPER NERD about etymology (and who doesn't?), "forte" comes from the feminine form of the French adjective of the same spelling and nearly identical meaning. In French, the masculine form (fort) is pronounced roughly "forr," while the feminine form (forte) is pronounced with no accent, as above (like the building). So even though French is all about the accents, this is one case where the common American pronunciation adds in a spoken accent that wasn't originally there! It would be written "forté" if it were supposed to have the "-ay" accent on the end!) Thus, while I will happily write the word "forte," I no longer use it in spoken conversation (except in discussing this very pronunciation issue!), lest I be drawn into the bottomless quagmire of pronouncing the word technically correctly and being thought of as an uneducated clod, or using the common pronunciation and betraying my nerdy, word-loving brain.
TL;DR it's officially "quicks-otic." English is weird.