No option to add Lambda functions in Amazon Lex intent under fulfillment

In the fulfillment section, Did you click Advanced options > Use a Lambda function for fulfillment?

If you did, then you only enabled the use of lambda function in that Amazon Lex Intent. It doesn't let you choose the Lambda function in that configuration.

You must do the following:

1. Click the "Save Intent" button.
2. Click "Build" button.
3. Click "Test" button when Build is done.
4. Click the Gear Icon on the Test chat that popped up and there you will see the Lambda functions.

You can also do it via AWS CLI: https://docs.aws.amazon.com/cli/latest/reference/lex-models/put-intent.html