Is there a way to query whether a subscriber has completed a certain journey and populate that subscriber information in a Data Extension in Marketing Cloud? Or any other way to retrieve this data??
You can use "Update Contact" Activity in your journey before exiting the journey and update the required data extension value.
You can also retrieve the data by going to the following path:
Email --> Tracking --> Journey Builder --> Click on the journey builder which you need.
In the summary box it will show you how many emails has been sent to user. Click on that hyperlink it will show the subscriber key ID's. You can also export the data.