Could you confirm what Dialogflow edition are you using? Is it Dialogflow ES edition or Dialogflow CX edition?
If you are experiencing occasional "Webhook call failed: DEADLINE_EXCEEDED" errors in your Webhook after long periods of inactivity, it may be undergoing a “cold start”. Here is an example for Cloud Functions (https://cloud.google.com/functions/docs/bestpractices/tips).
If this is the case, then consider the following solutions to address the issue:
Use an intent early in the conversation to "warm up" your services. Ideally the intent will have a simple message which you can define in both your services and the UI. If the Deadline Exceeded error occurs, Dialogflow will default to the response defined in the response section of your intent.
Use a custom web server as a webhook to achieve constant uptime and readiness.
Call or invoke your services periodically to reduce the chances of a cold start.
By default, webhook timeout is 5 seconds. However, in Dialogflow CX, you can go to Manage > Webhook, select the webhook you are using and increase the timeout to up to 30 seconds.
If you are using Dialogflow ES edition, you can extend the 5-second timeout limit up to 15 seconds by calling up to 2 follow-up events of intents one after another in your webhook. You can also check this documentation for more information.