Hi @jordan_webb , we’re glad you’re building integrations with our open API 
I believe the Backlog project’s webhook function will send information in JSON format about the created Backlog issue, to a specified URL endpoint which you can set in the webhook settings. See Add and Manage Webhook Settings
Then, at the URL endpoint (i.e. your server) where you receive the webhook (JSON data), you can program it to parse the JSON data in order to get the custom field information, before sending it to Zendesk.
Besides this, you can also use a service like Zapier to catch the webhook and parse the JSON data. For example:
Do check out these related posts:
- Integrate Backlog with external services via Webhook (and Zapier)
- Receive Backlog updates in Microsoft Teams using Zapier
Hope this helps!