Can gout turn into gangrene?

No. Gout is when uric acid crystals form in your joints. Gangrene is when you lose blood flow to your tissues and they die. Gout isn't associated with gangrene, but it wouldn't be impossible for someone with gout to also form gangrene.
NO. In gout the uric acid crystals will be deposited in the blood vessels causing severe systemic diseases but will not occlude completely to cause gangrene.