"Kwee-ree" (/kwɪə̯ri/~/kwɪri/~/kwiri/) is more common in dictionaries
The most common pronunciation of query seems to be "kwee-ree" (or more precisely, "kweer-y", since as Mazura says the vowel here is pronounced the same as a vowel before syllable-final "r", as in the words "fear" or "beard"; it's not necessarily pronounced the same as the "ee" sound of a word like "key-ring").
Looking it up in a bunch of online dictionaries using the OneLook Dictionary Search, I found that:
- all of the dictionaries with pronunciation guides list "kwee-ree" as either the only or the first pronunciation. More specifically, the IPA transcription of this vowel in dictionaries is either /kwɪər-/ (representing /kwɪə̯r-/, with shwa as a non-syllabic offglide; this may also be realized phonetically as a long [ɪː] that is contrasts in British English with the short [ɪ] sound in words like mirror or squirrel) or /kwɪr-/ (representing the usual American pronunciation resulting from a merger before /r/ of the "long e" vowel in words like serious with the "short i" vowel of words like Sirius; the neutralized vowel in this environment is traditionally identified with and transcribed as lax /ɪ/, but to some speakers it may sound more like tense /i/).
- only some dictionaries list "kweh-ree" (IPA /kwɛr-, kwer-/) at all: Merriam-Webster, Cambridge Academic Content Dictionary (click over to the "American" tab), Wiktionary, and Random House Kernerman Webster's College Dictionary 2010 (accessed via The Free Dictionary).
Modern dictionaries generally try to describe the language as it is used. They don't always list pronunciations strictly in order of how common they are, but I think this evidence considered together strongly suggests "kwee-ree" is more common. Another interesting detail, although the sample size may be too small to tell for sure: all of the dictionaries that did list "kweh-ree" were dictionaries of American or International English; none of them were of British English specifically.
"Kwee-ree" also seems more common out of 10 Youglish examples that I listed to
This is not a very scientific survey, but to a get a general sense of proportions, I listened to the first ten "Youglish" audio samples for query. Out of them, I counted 7 clear cases of "kweery," vs. 1 clear case of "kwerry" and 2 unclear cases that sounded a bit like "kwerry," but might have been extremely short versions of "kweery". So "kweery" seems to be more common, but "kwerry" is not all that rare.
The prescriptive perspective: also in favor of "kwee-ree"
You only asked which pronunciation is most commonly used, but I thought it might also be interesting to look at prescriptive guides to pronunciation such as Charles Harrington Elster's Big Book of Beastly Mispronunciations. Elster expresses a clear preference for "kwee-ree," which he says "is the preference of all authorities I have consulted from Walker (1791) to the present." According to Elster, the pronunciation with a short e "was first recorded in Webster 3 (1961)."
Side notes on etymology and possible pronunciation history
The source of the word is Latin quaere (and the word used to be spelled this way in English too). There seems to be some association (especially in British English) between the spelling ae and the "long e" sound: words like aesthete are sometimes pronounced in British English with the first vowel long, even though the general tendency is to use short vowels before consonant clusters that span more than one syllable, like -sth-. On the other hand, the Oxford English Dictionary does list one historical spelling of query that suggests a pronunciation with a short vowel: querries, which was apparently used in a Scottish text at some point before the 1700s.
I suspect the pronunciation with a short "e" may be due to analogy with "very," which is the only other word I can find that is spelled with "-ery" and has the stress on the "e." There may also be some general tendency for disyllables ending in unstressed -y (pronounced as /i/ or /ɪ/) to have short vowels in the first syllable: some other words that seem to exemplify this tendency are many, any, body, busy, dizzy, honey, study, copy, marry, carry.