Working solution is to add another decrement at the bottom of the outer loop:

using System.Collections;
using System.Collections.Generic;
using UnityEditor;
using UnityEditorInternal;
using UnityEngine;

[CustomEditor(typeof(DialogueTrigger))]
public class DialogueTriggerEditor : Editor
{
    private SerializedProperty _conversations;

    private void OnEnable()
    {
        _conversations = serializedObject.FindProperty("conversations");
    }

    public override void OnInspectorGUI()
    {
        //base.OnInspectorGUI();

        serializedObject.Update();

        _conversations.arraySize = EditorGUILayout.IntField("Conversations Size", _conversations.arraySize);

        for (int x = 0; x < _conversations.arraySize; x++)
        {
            var conversation = _conversations.GetArrayElementAtIndex(x);

            var conversationName = conversation.FindPropertyRelative("conversationName");

            EditorGUI.indentLevel++;
            EditorGUILayout.PropertyField(conversationName);

            EditorGUI.indentLevel++;
            var _dialogues = conversation.FindPropertyRelative("Dialogues");

            _dialogues.arraySize = EditorGUILayout.IntField("Dialogues size", _dialogues.arraySize);

            for (int i = 0; i < _dialogues.arraySize; i++)
            {
                var dialogue = _dialogues.GetArrayElementAtIndex(i);
                EditorGUI.indentLevel++;
                EditorGUILayout.PropertyField(dialogue, new GUIContent("Dialogue " + i), true);

                EditorGUI.indentLevel--;
            }

            if (_dialogues.arraySize > 0)
            {
                if (GUILayout.Button("Save Conversation"))
                {

                }
            }

            EditorGUI.indentLevel--;
            EditorGUI.indentLevel--;
        }

        serializedObject.ApplyModifiedProperties();
    }
}

Working solution is to add another decrement at the bottom of the outer loop:

using System.Collections;
using System.Collections.Generic;
using UnityEditor;
using UnityEditorInternal;
using UnityEngine;

[CustomEditor(typeof(DialogueTrigger))]
public class DialogueTriggerEditor : Editor
{
    private SerializedProperty _conversations;

    private void OnEnable()
    {
        _conversations = serializedObject.FindProperty("conversations");
    }

    public override void OnInspectorGUI()
    {
        //base.OnInspectorGUI();

        serializedObject.Update();

        _conversations.arraySize = EditorGUILayout.IntField("Conversations Size", _conversations.arraySize);

        for (int x = 0; x < _conversations.arraySize; x++)
        {
            var conversation = _conversations.GetArrayElementAtIndex(x);

            var conversationName = conversation.FindPropertyRelative("conversationName");

            EditorGUI.indentLevel++;
            EditorGUILayout.PropertyField(conversationName);

            EditorGUI.indentLevel++;
            var _dialogues = conversation.FindPropertyRelative("Dialogues");

            _dialogues.arraySize = EditorGUILayout.IntField("Dialogues size", _dialogues.arraySize);

            for (int i = 0; i < _dialogues.arraySize; i++)
            {
                var dialogue = _dialogues.GetArrayElementAtIndex(i);
                EditorGUI.indentLevel++;
                EditorGUILayout.PropertyField(dialogue, new GUIContent("Dialogue " + i), true);

                EditorGUI.indentLevel--;
            }

            if (_dialogues.arraySize > 0)
            {
                if (GUILayout.Button("Save Conversation"))
                {

                }
            }

            EditorGUI.indentLevel--;
            EditorGUI.indentLevel--;
        }

        serializedObject.ApplyModifiedProperties();
    }
}