By convention you make a POST request to a URI to create the new resource, this is not the same as tinker and Sequel Pro.
Make sure your app has a route definition which can respond to the request:
// routes/api.php
Route::post('users', 'UsersController@store');
In Postman, you change the method to POST and enter the URI http://your.domain.com/api/users and add a body which can be form data, raw application/json etc.; any of which will essentially be sending key-value pairs with the request.