I've installed contact storage module, it's provide storage for Contact messages which are fully-fledged entities in Drupal 8. What I want to do, is to insert data programmatically from my custom module into the entity of the contact storage
Yep. Just use any of the methods available to create the entity and save it.
use Drupal\contact\Entity\Message; $message = Message::create([ // And other fields required... 'subject' => 'Subject', 'message' => 'Message...', 'recipient' => 1, ]); $message->save();