To use REST API send email quick Action for cases, make sure that you have Email-to-Case enabled in your org.

Salesforce classic: Setup -> Customize -> Cases -> Email-to-Case

REST API URI: /services/data/v53.0/sobjects/Case/quickActions/SendEmail

HTTP Method: POST

Body:

{    

    "contextId": "500B00000082so8",

    "record": {

        "HtmlBody": "<div+style=\"font-family:+Arial;+font-size:+14px;\">Test</div>",

        "Subject": "Sfdcsupport",

        "ToIds": ["003B000000IqL5L"]

    }

}


Sample Request: 


Your output looks like below: Visit you case feed page, you can find the email attached over the case.