First of all it is important to know, that you need to apply for a travel document, you don't get it automatically. You can earliest apply when you receive your decision that you are granted refugee status from the Regional Asylum Office. If you got subsidiary protection, it will be more difficult to apply and only some people will be able to receive a travel document. You can not apply for a travel document, when you are still an asylum seeker. In general it is hard to say how long it will take until you receive your travel document, after you applied. This depends on the circumstances of your case, on the region of Greece where you applied and other factors.
After you applied the Greek police will carry out background and security checks to see, if you are allowed to receive a travel document. This can take many months, until these checks are completed, in some cases up to a year. If the decision is positive, you can apply for the actual travel document at the Greek Foreigners Police. This will again take between 2-4 months, until the travel document is ready and you can go and pick it up. On our website you can find detailed information about the process of getting a travel document here.