I’ve done some things with React, but I’m by no means an expert.
The way I see it, React and other libraries are there to help you with organization, not restrict you and confine you into certain design philosophies. With something small like the quote generator, there’s no reason you need to break it into smaller components.
Now, to answer your question, I’d probably just use one component too. I’d think about breaking off the actual quote/author into a second component, but it wouldn’t be too complicated to have them both defined in the master component.