Hi Augusto.
I'm wondering the basis for the claim that the Hungarian HG follows Arabic/Turkish modes? The tuning, aside from absolute pitch, is identical to the French instrument. The keyboard looks different (because of the spaces in it), and yes, one key looks inverted compared to a piano, but the lower row (the only row on some of the earliest examples known) plays the Ionian (major scale), and the upper row simply adds in the missing pitches on the Western diatonic scale (and no, it doesn't use any quarter tones). So I'm a bit confused about what you're trying to say. I know you play the Hungarian instrument, so you do know what you're talking about, and I suspect that I've misunderstood what you wanted to say, but without further explanation, your statement conveys something about the Hungarian instrument that isn't the case (that it uses a "different" scale from most European instruments).
Now it is true that Hungarian HG music uses a lot of modes besides Ionian, including Mixolydian (very common), Lydian (less common), Dorian (extremely common), and Aeolian (very common), all of which appear in the HG repertoire, but the modes one plays and the instrument's basic scale are two different things. Other modes, notably the Hungarian Gypsy scale (which does correspond to a maqam in the Kurd family), appear frequently in Hungarian music, but are not often found in the Hungarian HG repertoire. Maqāmāt consist of multiple ajnas, but the combinations are not fixed, so no single keyboard would be an "Arabic" keyboard.
However, if you assume an Ajam-family maqam, your statement would be generally true of both French and Hungarian HGs (as well as all diatonic major scale Western instruments), so that wouldn't seem to be what you mean as it wouldn't differentiate or tell us anything other than that there are Ajam maqāmāt that generally correspond to the western Ionian scale.
Regarding the bellydance repertoire, that I have less trouble seeing, since maqāmāt corresponding to the Hungarian modes (none of which are uncommon in Western folk music) exist and there was a considerable period of musical exchange in the region and ongoing contact with Balkan music where your statement about using Arabic modes is indubitably true.
Best,
-Arle