If I have a person stay with me, after surgery, am I supposed to pay them?

Since she volunteered, get her a very nice gift of thanks.Perhaps a gift card to a restaurant or a store.A gift card for gas and a car wash.Some nice slippers.

Unless you're very close friends, yes. However, the government usually subsidizers her time, anyway. Speak to her about it, and the relevant gov. department. 

If they have offered help and not mentioned payment then you are under no obligation or expectation of paying. However, if you can afford it, it might be appropriate to give a gift of money. These things can be hard to judge but my instinct would be to see how much help they actually provided after, say, 1 week and if it is  significant then put some notes in an envelope as a gift. And maybe again after another week. You need to think of it as a reciprocal gift rather than paying.

Having said all that, I'm in the UK and social rules are different in different countries.

I would pay her, maybe not in money but some service you can do for her in the future.

It's an awkward subject, when it comes ri remuneration for someone's act of kindness, but make an offer.

Volunteers are never paid. However you could make them coffee or maybe a cake?