This is true.
Also I know that J1 is usually thought of as a Semitic haplogroup whereas J2 is more associated with Anatolia which is likely the source of the J2 you find in Greece, Albania, Sicily and others. So the affinity with the Middle East is more with West, not Southwest Asia, which I think is what you're saying.
J2 seems to have arrived in Europe from Anatolia through the so called "Pelasgian peoples", that is an old Greek word used to call those people living in the northern mediterranean area before the indo-european migrations (J2 is found indeed in the whole Southern Europe, including Hiberia, the Balkans, Bulgaria, Romania, but the highest concentrations are in Greece, Portugal and Southern Italy).
It cannot be connected only to one people, even Etruscans belonged mostly to J2 (or at least did its aristocracy).
MODERN Anatolia and Arabia are mostly J1. J2 and J1 have a common stock but J2 shifted mostly in Europe (although it is still present in Middle East and among the high castas of India, the Brahmins).
Ethnically J2 can be connected to Minoics, Cretans, Etruscans, Pelasgians, Iberians, Indian Brahmins (as well as R1a).
Anyone of haplogroups J or E in Europe would trace it back to the Near East from the Neolithic.
Apparenly these haplogroups entered in Europe from Turkey in the neolithic. Yet ancient Turkey was very different from modern Turkey, becouse migrations from Central Asia and Europe had not happened yet). In Hiberia J2 holders found the local population holding R1b, in Italy J2 holders probably arrived for first, but later, with the arrivals of the Italics, the peninsula was invaded by R1b holders arriving from central Europe (indeed the R1b of Hiberia and Italy are a bit different). In Greece J2 was also older, followed, during the Doric times, by invasions of peoples arriving from the Balkans and holding R1b and R1a.
The haplogroup E entered first in the Balkans, where it has its highest concentration or at least it had in the past and from there it was spread in the Adriatic area, including Italy, Greece, Bulgaria and Romania. An other arrival happened in Spain.
Finally J2 arrived also in central Europe but with a far less intensity during the Neolithic era: peoples from Anatolia, crossing the Balkans, arrived in Western Germany.
This is what I know. Infos about this are spread here and there on the net.