Hi Kyle •,
Welcome to Microsoft Q&A forum.
As I understand, you want to know to ugrade Timescale DB version.
As per the information shared above and here in the documentation https://learn.microsoft.com/en-us/azure/postgresql/flexible-server/concepts-extensions, for Postgre 16 timescale db is N/A.
Could you please let us know to which version you are trying to migrate?
Thanks