There are two instruments in this recording.
In the left channel is a Greek bouzouki, which is a fretted instrument, tuned in typical Western fashion (12 tempered semitones).
In the right channel is an Arab Oud, which is a fretless instrument (imagine a medieval lute, but without the frets, played with a plastic pick that looks like a popsicle stick). I once had the chance of playing it for a few days, learning from a friend, a pro player from Syria.
Because one of the two instruments (the Bouzouki) has tempered tuning, the other guy (the Oud) is also by and large playing almost the same way (temperate) in this case. Conversely, the Bouzouki seems to try to imitate the quarter notes, occasionally, by bending the strings. (And it's not quarter notes, it's more like 30 cents off, but never mind that).
In other words, this is a very Westernized kind of Arab music. I wouldn't even call it Arab music, really. George Harrison plus a sitar does not Indian music make...
So this tune feels to me like a sort of a (pretty good, to be sure) crossover piece, with each instrument trying to accommodate for the other, creating some middle ground between the two traditions (Greek and Arab in this case).
A classic of this kind of thing is "A meeting by the River" with Ry Cooder (slide strat) and Vishva Mohan Bhatt (Indian classical musician who plays a slide guitar which is actually a sitar of sorts). Def check it out. I also once heard him (V.M. Bhatt) play Indian classical music live in Calcutta, and totally blew me away, but that's another story.
Anyway if you want to hear real Middle-Eastern, non-temperate scale based music, the above is not the right place to start, go for some real stand-alone Oud playing instead. And give yourself at least a few days of listening without trying to prematurely explain everything to yourself. Just listen and let it sink in by itself. As you are already a musician, it won't take long before your start to get it, and, quite possibly, enjoy it a lot too... :)