Well, to be fair, there are some venomous/poisonous animals in captivity lack some part of their natural habitat (usually diet) that is essential to toxin production, so there are precedents for someone unfamiliar with venomous snakes to wonder about them.
Poison dart frogs come to mind.
But yes, venomous snakes do not require any particular diet or anything like that to produce their venom, they are every bit as dangerous as their wild counterparts.
That is true for poisonous, but not venomous. There has not been one case of a animal loosing the toxicity of their venom from being captive, or diet. (except the obvious venomoid and unnatural procedures.