linkIds are only meaningful in the context of a Questionnaire.  Searching for linkId of "1.1" isn't terribly useful - you might get lots of hits, but it's going to mean something completely different in every Questionnaire.  We don't understand why anyone would want to search by this.  If you're searching for a standard code that corresponds to the question, that would be in either item.definition or item.code (and there are search criteria for both of those).  Might re-consider if there is a specific use-case with justification. Querying by answer within QuestionnaireResponse is generally unsafe because the meaning of questions is influenced by parent answer.  In general, if you're interested in searching by the data in a QuestionnaireResponse is to extract the data into other resources.  QuestionnaireResponses are intended to be retrieved by their metadata, not their data.