Q&A

 Journeyman

Warranty would possibly cover if it broke before the service interval.

 Master

With respect:
Speaking as someone who has actually managed a Hyundai dealer parts and service department and who has been the warranty administrator and submitted claims as well -- the timing belt is a wear item and is not a covered component unless you can prove manufacturers defect...which is impossible at 5 years.

If you could prove a timing belt tensioner failed, however...and that part was covered by either the 10/100 powertrain warranty to the original owner...or an extended warranty -- then maybe you have something to claim.