Update:
Coming 2025Q1 (January release) NQ functionality includes the ability to share via Canvas Commons:

• Product Blog Post
• Commons Release Notes (Jan 2025)

Original Post:
Clearing up any confusion around this question:

• New Quizzes content can not be shared by Commons at this stage. As per the New Quizzes Hub Roadmap, this is currently being slated for completion Q2 2024 and Beyond the timeframe.
• QTI download of a Quiz question is available as per this release.

For further updates, I'd stay tuned to the Product Blog (filter by New Quizzes) and New Quizzes Hub area.