I listened to video response on YouTube where a brittish girl asked for criticism of her attempt to take on an american accent ... in one of the response a guy, "rubdaddy", responds he seems to be saying inquiry in a way I never heard before ... "inkwiri" instead of "inkwairy" (sorry for my improvised notations) .. when you look at the there´s a quite some variations, but none seems to correspond to the one "rubdaddy" is using. Is this related to his south accent or what do you think?

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he seems to be saying inquiry in a way I never heard before ... "inkwiri"
Hmm. I'd say he is saying inquiry in a way I've almost always heard in the U.S.

Is this related to his [Southern] accent or what do you think?
No. His Southern accent is quite mild. This is only Tennessee, not Alabama or Georgia! His pronunciation of inquiry can be heard anywhere in the States. There's nothing about that pronunciation that would draw the slightest bit of attention from anybody here.

CJ
Ok, thanks Jim. I guess I need to listen again. Still the one in the dictionary is different, isn't it? Or he's just talking fast.

(Also I noticed I had like hundred grammar errors in the my question posted - didn't reckon it was me writing that crap).

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It's one of the pronunciations shown in the dictionary.
There are alternate pronunciations, but I would say he used the one that's most common in the U.S.

CJ
thanks, hence the one with the schwa sound (ĕn'kwə-rē).