When did Dean Martin get a glass eye?

Answer 1

He didn't. You may be thinking of Dean's friend, and fellow Rat-Packer, Sammy Davis, Jr.

Also, in the movie The Sons of Katie Elder (1965), he carried a glass eye as a conversation piece. In one scene he pretended the glass eye was actually needed but in need of money was willing to raffle it to raise some money. After the winner of the raffle claimed the eye, Dean Martin's character offered to buy it back for less than the amount of the raffle proceeds, claiming he would feel bad not having it. The winner relented and sold it back to him. When Dean Martin's character started to put the glass eye in his pocket, the winner of the raffle asked him if wasn't going to put it in his eye. Dean Martin asked, "Don't you think it will be awfully crowded?" and raised his eye patch to reveal he still had a good eye.

Answer 2

In one of his comic strips a man and his opthamologist meet on the street and the doctor says "How is that glass eye I fitted you with?" then slaps his patient very hard on the back and his good eye pops out. That is what I remember.