Atlassian provides a Jira/QC synchronizer tool, for a price.
https://marketplace.atlassian.com/plugins/hpsync-plugin.xml
https://marketplace.atlassian.com/plugins/com.go2group.jira.plugin.mercury_kit
HP does not provide a synchronizer specifically for Jira, but you could try to build your own using the ALM Synchronizer SDK.
https://hpln.hp.com/group/synchronizer-content-alm
https://marketplace.atlassian.com/plugins/hpsync-plugin.xml
https://marketplace.atlassian.com/plugins/com.go2group.jira.plugin.mercury_kit
HP does not provide a synchronizer specifically for Jira, but you could try to build your own using the ALM Synchronizer SDK.
https://hpln.hp.com/group/synchronizer-content-alm