### Apache Airflow Provider(s)

snowflake

### Versions of Apache Airflow Providers

`apache-airflow-providers-snowflake==4.0.2`

### Apache Airflow version

2.5.1

### Operating System

Debian GNU/Linux 11 (bullseye)

### Deployment

Official Apache Airflow Helm Chart

### Deployment details

_No response_

### What happened

After upgrading the _apache-airflow-providers-snowflake_ from version **3.3.0** to **4.0.2**, the SnowflakeCheckOperator tasks starts to throw the following error:

```
  File "/home/airflow/.local/lib/python3.9/site-packages/airflow/providers/common/sql/operators/sql.py", line 179, in get_db_hook
    return self._hook
  File "/usr/local/lib/python3.9/functools.py", line 993, in __get__
    val = self.func(instance)
  File "/home/airflow/.local/lib/python3.9/site-packages/airflow/providers/common/sql/operators/sql.py", line 141, in _hook
    conn = BaseHook.get_connection(self.conn_id)
  File "/home/airflow/.local/lib/python3.9/site-packages/airflow/hooks/base.py", line 72, in get_connection
    conn = Connection.get_connection_from_secrets(conn_id)
  File "/home/airflow/.local/lib/python3.9/site-packages/airflow/models/connection.py", line 435, in get_connection_from_secrets
    raise AirflowNotFoundException(f"The conn_id `{conn_id}` isn't defined")
airflow.exceptions.AirflowNotFoundException: The conn_id `None` isn't defined
```

### What you think should happen instead

_No response_

### How to reproduce

- Define a _Snowflake_ Connection with the name **snowflake_default**
- Create a Task similar to this:
```
  my_task = SnowflakeCheckOperator(
      task_id='my_task',
      warehouse='warehouse',
      database='database',
      schema='schema',
      role='role',
      sql='select 1 from my_table'
  )
```
- Run and check the error.

### Anything else

We can workaround this by adding the conn_id='snowflake_default' to the SnowflakeCheckOperator.

### Are you willing to submit PR?

- [X] Yes I am willing to submit a PR!

### Code of Conduct

- [X] I agree to follow this project's [Code of Conduct](https://github.com/apache/airflow/blob/main/CODE_OF_CONDUCT.md)

super().__init__(conn_id=snowflake_conn_id, **kwargs)
        warnings.warn(
            """This class is deprecated.
            Please use `airflow.providers.common.sql.operators.sql.SQLExecuteQueryOperator`.
            Also, you can provide `hook_params={'warehouse': <warehouse>, 'database': <database>,
            'role': <role>, 'schema': <schema>, 'authenticator': <authenticator>,
            'session_parameters': <session_parameters>}`.""",
            DeprecationWarning,
            stacklevel=2,
        )
    def _process_output(self, results: list[Any], descriptions: list[Sequence[Sequence] | None]) -> list[Any]:
        validated_descriptions: list[Sequence[Sequence]] = []
        for idx, description in enumerate(descriptions):
            if not description:
                raise RuntimeError(
                    f"The query did not return descriptions of the cursor for query number {idx}. "
                    "Cannot return values in a form of dictionary for that query."
                )
            validated_descriptions.append(description)
        returned_results = []
        for result_id, result_list in enumerate(results):
            current_processed_result = []
            for row in result_list:
                dict_result: dict[Any, Any] = {}
                for idx, description in enumerate(validated_descriptions[result_id]):
                    dict_result[description[0]] = row[idx]
                current_processed_result.append(dict_result)
            returned_results.append(current_processed_result)
        return returned_results
class SnowflakeCheckOperator(SQLCheckOperator):