Push Array Inside Object in Angular

You're doing this the wrong way. Either you'll have to create a conversations FormArray to your parent FormGroup and then push FormGroups to it. Or you'll have to add the conversations control using the addControl method on a FormGroup.

I'd recommend using the first approach. So here's how you can do it following that:

import { Component } from '@angular/core';
import { FormBuilder, FormArray, Validators, FormGroup } from '@angular/forms';

@Component({
  selector: 'my-app',
  templateUrl: './app.component.html',
  styleUrls: ['./app.component.css']
})
export class AppComponent {

  itemsForm: FormGroup;
  selectConversionsForm: FormGroup;

  constructor(private fb: FormBuilder) { }

  ngOnInit() {
    this.itemsForm = this.fb.group({
      name: [null, Validators.required],
      description: [null, Validators.required],
      conversions: this.fb.array([])
    });

    // Seed selectConversionRows form group with the seed data
    // Generally this would come from the user selection. I'm assuming that you're already getting that.
    const conversions = [
      { conversion_id: 1 }, 
      { conversion_id: 2 }, 
      { conversion_id: 3 }
    ];

    // Seeding the Conversions   
    this.selectConversionsForm = this.fb.group({
      selectConversionRows: this.fb.array(conversions.map(conv => this.generateConversionFormArray(conv.conversion_id)))
    });

    const selectedConversions = this.selectConversionRows.value;
    selectedConversions.forEach(conversion => this.conversionsArray.push(this.generateConversionFormArray(conversion.conversion_id)));

    // EXPECTED RESULT: 
    //   name: 'sample name', 
    //   description: 'sample desc', 
    //   conversions: [{conversion_id: 1}, {conversion_id: 2}, {conversion_id: 3}]
    console.log(this.itemsForm.value);
  }

  generateConversionFormArray(id) {
    return this.fb.group({
      conversion_id: this.fb.control(id)
    });
  }

  get conversionsArray() {
    return (<FormArray>this.itemsForm.get('conversions'));
  }

  get selectConversionRows() {
    return (<FormArray>this.selectConversionsForm.get('selectConversionRows'));
  }

}

Here's a Working Sample StackBlitz for your ref.

UPDATE:

Here's how you would do it in your code:

import { Component } from '@angular/core';
import { FormBuilder, FormArray, Validators, FormGroup } from '@angular/forms';

@Component({...})
export class AppComponent {

  itemsForm: FormGroup;
  selectConversionsForm: FormGroup;

  constructor(private fb: FormBuilder) {
    this.itemsForm = this.fb.group({
      name: [null, Validators.required],
      description: [null, Validators.required]
    });

    this.selectConversionsForm = this.fb.group({
      selectConversionRows: this.fb.array([])
    });
  }

  ngOnInit() {
    this.onCreateItem();
  }

  onCreateItem() {
    const conversions = [];

    (this.selectConversionsForm.get('selectConversionRows') as FormArray).getRawValue().forEach(item => {
      conversions.push({
        conversion_id: item['conversion_id'],
      });
    });

    this.itemsForm.addControl('conversions', this.fb.array(conversions.map(conv => this.fb.group({
      conversion_id: this.fb.control(conv.conversion_id)
    }))));
    console.log(this.itemsForm.value);
  }

}

PS: Do Read the Comments in the Code to understand more.