Hi Peter, I was able to do this quite successfully, however needed Apptio's guided services to create new "Issue types" been spikes and defects, once done they also connected the two with JS mappings. Just bear in mind your hierarchy needs to be aligned across both systems and that teams must use a consistent "Parent link" in JIRA so that TP can pick this up parent link and add is as a entity under a Story.
I dont think this is out of the box.
#Targetprocess