Water is no-toxic and yet lethal. So’s a rope.
You're, right now, breathing a gas mixture which is 78% nitrogen. Nitrogen, in and of itself, does nothing to harm you. It does not disrupt your physiological functions. Hence, it is not toxic.
You're, right now, breathing a gas mixture which is 19% oxygen. That oxygen is required for your physiological functions to occur. If you are deprived of that oxygen, your physiological functions will cease: you will die.
Breathing 100% nitrogen deprives you of the necessary 19% oxygen, causing death. Hence, it is lethal.
I can see why you think so, but it is exactly true. It is not the nitrogen that kills you. It is the lack of oxygen. The reason why it is so effective is that nitrogen does not trigger "I'm suffocating" response in the body. Excess CO2 buildup is what makes you gasp for air. In industrial settings where you can have excessive concentrations of nitrogen in the air where workers may be, you have to follow a lot of safety protocols. This is because a worker can go into a room with a high concentration of nitrogen, and he doesn't even realize he's dying until he's already passing out.
That is why it is the perfect thing for executions. If you don't tell the condemned when the concentration starts to increase, he simply passes out and dies without ever realizing it happened.