No, exporting and downloading a combined recording, such as a Clip or the All Participants Backup, does not use your account's available track download hours.
Only downloads of separate High Quality Tracks and integrations that include those tracks (like exporting to Descript or as an Adobe Premiere Pro timeline) count toward your hours.
NOTES:
Download hours vary based on your Riverside subscription plan.
You can create Clips and full recordings in Riverside's Editor and export them without any limit.