Yes. But in the language of metaphysics, person and nature are not the same thing. Person is an identity. "George W. Bush" is a person. That's his identity, his "who". His what is a nature. It's the collection of qualities and capabilities that are proper to the class to which he belongs, that of human beings.
God is three persons with one divine nature. You know about things that have nature without person; a rock has a rock nature but there is no "rock person", so a rock exhibits a zero-to-one relationship of person to nature. You, yourself, have a human nature, and are a human person; that's a one-to-one relationship of person to nature.
God has a three-to-one relationship of person to nature. This is without precedent in the created world, which has only one-to-one (humans and angels) and zero-to-one (inanimate objections, animals) relationships of person-to-nature. But it is not illogical, or even paradoxical.