Why didn’t Turkey claim any Greeks island near their shores?

They did: Imbros and Tenedos. Like the other islands, they were substantially ethnic Greek, but they remained in Turkey under the Treaty of Lausanne, presumably because of their strategic importance outside the Dardanelles.

Of the other Aegean islands near the shores of Turkey, the islands from Samos up were ceded to Greece by the Ottoman Empire in the aftermath of the First Balkan War; the Dodecanese went to Italy instead, and were integrated into Greece in 1946.

Fine print in the treaty around the Dodecanese led to the Imia/Kardak sovereignty crisis in 1996. The West Wing made fun of the incident a few years later, situating it on the Greek/Albanian border. I was in Greece at the time, and it was not effing funny.

As for claiming, well, Turkey lost the Balkan War, hence losing most of the islands, and won the Turkish War of Independence, thus getting to claim two strategic islands. Right and wrong doesn’t enter into it, and nor does self-determination; but FWIW, the population of the islands was substantially ethnic Greek even before the population exchanges—although a Turkish minority has remained in place in the Dodecanese, which were not subject to the exchanges.