If your pet needs euthanized as the result of an injury or illness that is covered under the pet insurance then yes most pet insurance will cover the cost of having the pet euthanized.
Some pet insurance covers the cost of euthanasia while some other pet insurance policies don't cover the cost of euthanasia.
Also if the pet has a pre existing condition that causes the pet to need to be put down then the pet insurance will not cover the cost of pet euthanasia or cover the pre existing condition.