To interact with your conversational agent you have to use Dialogflow CX libraries. Actually, Dialogflow CX is being deprecated tomorrow October 31 in favor of these conversational agents, and if you see the conversational agent UI is the same as Dialogflow CX but expanded with more generative AI functionalities such as playbooks.

The method to interact with an agent by text is called "detect intent". You have a node.js sample here: https://cloud.google.com/dialogflow/cx/docs/quick/api#detect-intent-nodejs

But you need to make sure that you are authenticating properly when you do the call to the API. That means that if you are executing the call from your local machine, you have to be logged in gcloud as a user with the corresponding permissions or use a service account with those permissions and add the google credentials in the request.