There are few ways to approach this if you are doing a RESTful integration and need to send a JSON payload to an external service. Sometimes, you want to go the custom code route, and the general approach you can use is there to create a wrapper class for the request like the following:
 

public class SomeJSONrequestWrapper { public string record_id { get; set; } }

And then use JSON Class serialize method (https://developer.salesforce.com/docs/atlas.en-us.apexref.meta/apexref/apex_class_System_Json.htm#apex_System_Json_serialize) to create your JSON payload.
The JSON.serialize method will convert the data in your wrapper class to the corresponding JSON format.

You can also do something similar for the response except you'll deserialize (https://developer.salesforce.com/docs/atlas.en-us.apexref.meta/apexref/apex_class_System_Json.htm#apex_System_Json_deserialize) the response into your response wrapper.  You need to make sure that your wrapper class match the JSON structures very closely.  Then you'll end up with a Salesforce object that's been deserialized from the JSON structure.  Once it's an object, then you can work with it in Salesforce APEX code.

You'll also have to make some type of HTTP request like the following example for a POST request:

Http h = new Http(); 
HttpRequest req = new HttpRequest(); req.setHeader('Accept', 'application/json'); 
req.setHeader('Content-Type', 'application/json'); 
req.setEndpoint(url); 
req.setMethod('POST'); 
req.setTimeout(10000); 
req.setBody(someBody); 
HttpResponse res; res = h.send(req);

Typically, you will call your API/webhook methods asynchronously within an future method called from an Apex trigger or from batch context.

There are also some AppExchange apps which can make this way easier and don't require any code like Declarative Webhooks (https://appexchange.salesforce.com/appxListingDetail?listingId=a0N3u00000MSv8REAT).