It's not exclusive from Nidus. I can't remember if Nidus has it as a passive or if he has to sit in the Helminth chair to start the infection but regardless.
You get it from bumping/contacting another player in your Squad who is infected. Or you can get Nidus and infect yourself in the chair.
A small visual effect and sound can be heard when you get infected, but it can be easy to miss. You will notice a small cyst will appear on your Warframes neck (visibility differs between Warframes but it's always located in the same spot) from then, you need to wait a full week (doesn't matter if you login or not for that week) for the Cyst to fully mature.
Once fully matured you can use it to make the Helminth pet, or alternatively. If you haven't unlocked the Helminth room yet. The door to the Helminth room will open when approaching it with a matured cyst. You can interact with the chair and it will remove the cyst.
Be aware, if you choose to remove a Cyst from a Warframe, that Warframe will become immune to it and cannot be infected again. (meaning you need to use a different frame to get infected) Most players will probably choose to remove it since it messes with fashion frame. I think Nidus is the exception, able to reinfect himself and remove it over and over.
Do note the Immunity only applies to the Warframe that got infected, meaning if you have two default Excaliburs, you can cure one, and the other can still be infected.
EDIT: I forget how you unlock the Helminth room permanently but be aware once you cure a Warframe, they cannot enter the room again. So if you feel like sightseering make the most of it before you exit the room again.
Zahnrad's post in need advice on helminth cyst acquisition was marked as the answer