There's no native support for index management. Search indexes are considered downstream data structures, accepting content from other data sources that collect operational data. As such, there's no built-in support for backing up and restoring indexes because the expectation is that you would rebuild an index from source data if you deleted it, or wanted to move it.
However, if you want to move an index between search services, you can try the index-backup-restore sample code in this Azure Cognitive Search .NET sample repo.