I watched many YouTube videos and they seem to pronounce the first part of "Quarter" exactly the same as "core". It seems that the A is silent reading. Am I correct? Are there any accents that pronounce it like "QuAr"?
The A always sounds like an O. I never hear it pronounced like an A.
It's the U that you hear disappearing in regular everyday speech. Sometimes it's briefly pronounced, but often not at all. You can see this described in this YouTube video. You might hear the U if someone specifically emphasizes that word in a sentence.
I'm a native English speaker from Canada. I can't always speak about British English, but this seems to be true there too.