If somebody can get as far as recognizing the problem as requiring Dijkstra's algorithm, they can go pull a book off the shelf, or use Google, and find the details of the implementation. Or, more likely, find a pre-existing implementation[1] that can be re-used.

Personally, I doubt that many people are walking around with the deep details of Dijkstra's memorized and readily available, unless they have a job that very specifically focuses on graph operations and they do nothing else day in and day out. So you could memorize it just for the interview, and then forget it again by the next day. But is there really any point to all that?

[1]: https://www.google.com/search?num=30&q=graph+algorithm+libra...