It was suggested to me that one might determine the version of a Questionnaire by resolving the meta.profile StructureDefinition URL and reading the FHIR version from that.  I tested that, and got the wrong FHIR version for an STU3 URL.  Note:  The browser seems to request the HTML page, and that works.

I used "curl" to resolve the URLs from the command line.  For example:

curl http://hl7.org/fhir/3.0.2/StructureDefinition/Questionnaire
  returns a redirect to http://hl7.org/fhir/STU3/StructureDefinition/Questionnaire,
which redirects to http://hl7.org/fhir/StructureDefinition-Questionnaire.xml,
which returns the XML StructureDefinition, but in R4.