I pronounce the word in the title as ˈkwɔːtə. However, I have been aware for many years that some people pronounce it as ˈkɔːtə, because my wife does this. The same is true of all words containing <quart>. John Wells gives both pronunciations in LPD, but with the kw form first. This is also true of the Heinemann New Zealand Dictionary 1979. For Gen Am Wells gives only the kw form and this is supported by A Pronouncing Dictionary of American English (Kenyon & Knott, 1953). EPD14 doesn’t mention the w-less form at all. CEPD 2003, however, does list both pronunciations.
It has only just occurred to me to ask if the w can go missing in other words like Quorn, quorum, squaw and squawk. It appears not. My wife has w in all these words and all the dictionaries I have consulted only list kw forms.
I must confess I can’t remember hearing any Brit speakers use the w-less forms, but perhaps I haven’t been listening hard enough.