Begin by exporting your data from Coda. Coda allows you to export documents in various formats such as CSV or JSON. Navigate to the document you wish to export, click on the 'Share' or 'Export' option, and select the desired format. For this guide, exporting as CSV is recommended for simplicity.

Prepare your Apache Iceberg environment. Ensure you have a compatible compute engine like Apache Spark, Flink, or a compatible SQL engine with Apache Iceberg support. Install the necessary Iceberg libraries and dependencies for your chosen engine to enable you to create and manage Iceberg tables.

Define the schema for your Iceberg table. This schema should correspond to the structure of your data exported from Coda. Use your compute engine to execute a SQL statement that creates a new Iceberg table. For example, using Spark SQL:
```sql
CREATE TABLE iceberg_db.my_table (
id INT,
name STRING,
value DOUBLE
)
USING iceberg;
```

Convert your exported CSV data into a format that can be ingested by your compute engine. If necessary, clean or transform the data to match the schema defined in your Iceberg table. This might involve using simple scripting in Python or a similar language to ensure data types and formats align.

Load the prepared data into the Iceberg table. Use your compute engine's native capabilities to read the CSV file and write the data to the Iceberg table. For example, using Spark:
```python
from pyspark.sql import SparkSession

spark = SparkSession.builder \
.appName("Coda to Iceberg") \
.getOrCreate()

# Read CSV data
df = spark.read.csv("path/to/exported_data.csv", header=True, inferSchema=True)

# Write to Iceberg table
df.write \
.format("iceberg") \
.mode("append") \
.save("iceberg_db.my_table")
```

After loading, verify that the data has been correctly ingested into the Iceberg table. Run queries to check the data integrity, ensuring that the number of records and the data values are correct. Use simple SQL queries through your compute engine to perform these checks:
```sql
SELECT COUNT(*) FROM iceberg_db.my_table;
```

Finally, optimize and manage your Iceberg table for performance and storage efficiency. Execute maintenance operations like compaction and partitioning if necessary. Use Iceberg's built-in commands to optimize the table:
```sql
CALL iceberg.system.rewrite_data_files('iceberg_db.my_table');
```

By following these steps, you can successfully move data from Coda to Apache Iceberg without relying on third-party connectors or integrations, using the built-in capabilities of your chosen compute engine and Apache Iceberg.