The easiest way to achieve this would probably be the Jenkins Generic Webhook Trigger Plugin.
Once you have configured an inbound webhook trigger in Jenkins, you need to invoke it somehow. As is, the various curl based examples there could all be executed via Bamboo's Script task.
If you are wondering why a webhook is not yet amongst the supported Bamboo notification recipient types, you are not alone - please watch and vote for the following issue to further increase Atlassian's priority for this long missing and highly requested feature:
If you happen to use AWS, you could also work around the missing webhook notification type via the flexible Automate with AWS notification recipient in our (commercial) Automation with AWS (Bamboo) app, which offers various remote actions such as Invoke Lambda Function, which could then trigger the Jenkins webhook in turn.
Hey there, Data Center community! I'm Martyna Wojtas and I am the Product Manager for Bamboo Data Center. I’m excited to share that Bamboo 9.0 is now available. We purpose-built this platform to help...