Yes, you can do it with Contextual filters in Views. You can Duplicate the Taxonomy term Views and work on it, disable the original for future reference.
You need to add one Contextual filter for each of your Vocabularies. You probably have one field for each of those on your Content type, so add the Taxonomy reference fields as your Contextual filters. The alternative is to use "Has taxonomy term" twice and limit Vocabularies.
You'll need to set up a default value taken from the URL. You can find a lot of tutorials online about the details of setting it up: Add a contextual filter to a view, How to use Contextual Filter in Drupal 8.
The above approach will only work with term IDs
To have term names displayed instead of TIDs you need to add a Relationship to the Taxonomy field to your Views, then add
Taxonomy term: Name in your Contextual filters and it will use that Relationship in the Taxonomy Contextual filter. Repeat for the other Vocabulary. Also see this answer: How to setup view to filter based on second URL parameter?
That is how you can get