Like the other fellas said, it's really it's own. So no like Ghost said and in-between like SC said.
It's a bit more complex than that because the Arabic language withing each dialect has it's own uniqueness. For example, there is classical Arabic which is more common to all Arab-speaking people. Then there's formal Arabic which has similarities between all Arabic speaking countries & people, but is also unique to each country and lastly there is colloquial Arabic. Basically the non-formal, uniquely inherent to each people and that colloquial Arabic that is spoken in Egypt is about as unique as can be. You won't hear anything close to it in any of the other Arabic-speaking countries or people. For example, if you hear Egyptians speaking that regular, homestyle street and buddy-buddy talk, you'll recognize it instantly because no one else in the Arab world speaks it that way. It's almost uniquely slangish. What you're hearing in those leaked secrete recordings is the typical Egyptian colloquial but mixed with some formal Arabic. If you listen to Egyptians on the streets, that's the pure colloquial Egyptian Arabic that doesn't exist anywhere else.
There are actually other accents within Egypt and Egyptian Arabic such as Asyuti Arabic spoken in certain areas of the northern Nile Delta region as well as Bedouin Arabic which has certain Berberis dialect similarity to it and if there is any Arabic with Maghrebi flare, that's the one.
Palestinian Arabic is closer and part of Sham or shamy Arabic, spoken essentially in El Sham regions such Syria, Lebanon & Jordan. That's closer to Egyptian formal Arabic. Iraqi Arabic is closer to GCC Arabic which is about as traditional as it gets since that's really the birthplace of the language.
So as you can see, it's a mumbalumbgum clustermush.
Well, TBH how could anyone of us know what dialect was spoken during "ancient" Quraish? 'Ancient' being the operative word since even if the language was passed down through the generations and spoken in the same manner, it would be closer to Saudi Arabian, GCC & Yemeni Arabic than anywhere near Palestine. I think your friend has an honest, loyal bias to Palestinians which is perfectly fine and might feel like giving them an honor as such, but it's inaccurate. If anything, Palestinian Arabic is closer to Shamy Arabic like Ghost was saying.