We live in AZ and have a rental home in TX. Home was seen my rental management on Monday, key given to new tenant on Monday night to start moving in. Tuesday tenant states garage door is damaged. Two upper panels of garage door are pushed out from interior to exterior and damaged beyond repair per garage door company. Management company states "that in the short period of time that the tenants had possession of the property and what took place with the garage door it is ultimately the property owners responsibility". Is this correct?

No, yes and maybe, unless you can prove that the new tenant did it, you will have to fix it.

Was there a walk though with the new tenant by the management company, did they inspect to show the home was OK on Monday night

So unless you can prove the tenant did it, you have little recourse