For integrating Amazon MQ with API Gateway, you will not be able to use the "AWS Service" integration. The reason is, "AWS Service" integration is only helpful if the AWS MQ Management API expose an operation to send messages to a MQ Broker. But if you look at the operations exposed by MQ API [1] they just offer the management operations. This is the reason, you cannot use "AWS Service" integration for your use case.
So if your API needs to connect to a MQ Broker and send message, I would suggest your API need to use "Lambda function" integration. To understand more about this, I would recommend to use the following documentation [2].
Then you can implement a Lambda function that initializes a message producer and send the message. For specific source code to implement the message producer and send the message please refer to the section titled "Step 2: Connect a Java Application to Your Broker" in [3].
[1] https://docs.aws.amazon.com/cli/latest/reference/mq/index.html
[2] https://docs.aws.amazon.com/apigateway/latest/developerguide/api-gateway-create-api-as-simple-proxy-for-lambda.html
[3] https://docs.aws.amazon.com/amazon-mq/latest/developer-guide/amazon-mq-getting-started.html#connect-java-application