Airbyte is an open-source data integration engine that helps you consolidate your data in your data warehouses, lakes and databases.

Airbyte

How can I save Airbyte data, so I can control version or deploy to server?

I am guessing when you setup DBT, you could also configure the snapshot feature to version data from Airbyte manually:
<https://docs.getdbt.com/docs/building-a-dbt-project/snapshots/>

(maybe you could expect these sort of setup better integrated in airbyte and accessible from a click in the UI...)