if I married a Chinese citizen in New Zealand (where I'm from), could I then apply for a Q visa?

It's possible. Yes, you should submit your application to Chinese consulate general that has jurisdiction in your area. If you can't submit your application on your own, you can ask your husband or friend to submit on your behalf.

Except for your marriage certificate, you should provide your husband's Chinese ID card (copy verision), your passport, application form and a recent passport photo.