Even though Qantas does not offer credit cards to the U.S. residents, you can become a member of their frequent flyer program. The membership will allow you to earn miles and redeem them with Qantas. To earn some more miles, you can apply for a travel rewards credit card. Such cards allow earning miles on all purchases, including on airfare, and they allow redeeming earned miles with any airline of your choice, including with Qantas.
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All rates and fees, and other terms and conditions of the products mentioned in this article/post are actual as of the last update date but are subject to change. See the current products' Terms & Conditions on the issuing banks' websites.