The information is correct. Looking at the majority of Greco-Turkish encounters in history and modern politics, people tend to believe that Turks always hated Greeks and vice versa. An important fact to remember is that the two countries enjoyed an impressive degree of cooperation following the Turkish War of Independence (1919-1923), until the Cyprus Crisis in the 1950s. Having beaten the Greeks in the War of Independence, the Turkish leadership decided not to press further. A Greece in tatters could certainly yield some benefits to Turkey; in the form of some touristic islands and in incorporating the Turkish-majority West Thrace region, for instance. But it would also draw new, more dangerous and ambitious powers to the region, Italy and Bulgaria. Turkish support to Greece is, therefore, inseparably linked to the country's feeling threatened by the revisionism of Italians and their Bulgarian allies (the king of whom, Boris, was married to the daughter of the Italian king in 1930). Also linked to this perceived threat was the formation of the Balkan Entente, a defensive alliance, in 1934 by Turkey, Greece, Romania, and Yugoslavia. That there were (quite probably accurate) reports that Bulgarians wanted to redraw the Turkish-Bulgarian border in Thrace to their advantage did not help. See this Twitter link for a nice drawing from 1935 from a Turkish newspaper. The female figure looking threateningly represents Turkey protecting her national borders and the Turkish inscription below reads "You want Thrace, there it is. Come and take it!". A joint Italo-Bulgarian invasion of Greece in 1940 (or before, for that matter) would clearly back Turkey to the corner and the latter did what it could to prevent that from happening.

Now, at any rate, the Turkish attitude to the Italians and fascism in the 1930s is a very complicated matter. So complicated as to deserve a separate PhD subject and luckily there is me who is doing just that! In short, Turks appreciated some internal aspects of the fascist regime. Following the economic crisis of 1929 and the following crisis of democracy in the West, most Turks believed a strong leadership with effective methods of propaganda was the way to go. Yet, they were almost unequivocally clear that (especially from the mid-30s onwards, it seems) Italian foreign policy was detrimental to Turkish interests. As before, this was related to the Bulgarian demands. It also had links to a much more proactive Italian policy in the Mediterranean region and also in Ethiopia which Turkey also found unnecessarily aggressive. So, in this sense, Greece was only one front of the clash of interests, as it were.

Finally, I have explained in a different post how Turkish leadership at the time sought peace whenever possible. This, I think, was due to their belief that the country was left "backwards" because of prolonged wars which drained the country's personnel and economic resources. The Italo-Bulgarian alliance represented a threat to this policy and, thus, the Turks decided to throw their support behind the Greeks.