We are migrating attachments from CQ to JIRA. Probably it was done many time in the past.
Can somebody, who done it, suggest on the easiest way of doing it or/and share the method?
If CQ stands for ClearQuest then I think that it should be possible by CSV.
Here is a documentation of how to prepare a valid CSV file with section about Importing attachments - https://confluence.atlassian.com/adminjiraserver073/importing-data-from-csv-861253680.html#ImportingdatafromCSV-formatPreparingyourCSVfile
Basically those should be available and you define the path in a URL form that would get the attachment and load them to Jira.
URLs for attachments support the HTTP and HTTPS protocols and can be any location that your JIRA server must be able to access. You can also use the FILE protocol to access files in the
import/attachmentssubdirectory of your JIRA home directory.