No, you cannot fine-tune Codex models like code-davinvi-002 using the OpenAI API. Fine-tuning is currently only supported for models such as gpt-3.5-turbo.
For coding tasks, OpenAI recommends using GPT-4 or GPT-3.5 with system instructions or examples (few-shot learning) instead.
Here OpenAI codex guide- https://oragetechnologies.com/openai-codex/