Which sentence is more common and natural to use?
Can you do me a favor?
Can you do this favor for me?

Question

Answer

both

Answer

Both are correct. With the second sentence, you have a more specific thing that you want help with. The first question is a more open question.

Answer

"Could" is more polite.  "Can" is used with family and friends (but not usually coworkers).

"Could you do me a favor?" is usually followed by some details.  Example: "Could you do me a favor?  I need a ride to the airport on Monday."

"Could you do this favor for me?" usually follows the description of "this" favor.  "I need a ride to the airport on Monday.  Could you do this (favor) for me?"  Often, "favor" is omitted.