Yes, but probably not as fluently as the show makes out. Crozier had been to the Arctic three times before. Especially during his first expedition, on the Hecla with Parry, this involved some extensive and relatively close contact with the Inuit. This was in Igloolik and Winter Island, in the Eastern Arctic. The Inuit there would have spoken a different dialect than the Netsilik, but they are both dialects of Inuktitut.

Some Inuit accounts of the Franklin expedition mention white men who knew some Inuktitut, like the word for "seal". Other accounts mention talking with white men, perhaps different individuals, or involve information that was probably learned from talking with them (I don't have my copy of Unravelling the Franklin Mystery with me, but I remember Woodman discussing an Inuk man who knew of Crozier's previous experience in the Arctic, which, according to his analysis, must have learned this from Crozier himself). That said, he probably didn't speak Inuktitut as well as the show depicts. Inuktitut is a difficult language to learn, and none of the explorers of this period mastered it.

Sources:

Unravelling the Franklin Mystery: Inuit Testimony, David C. Woodman

Second in Command: A Biography of Captain Francis Crozier, R.N., F.R.S., F.R.A.S., May Fluhmann