"I would like to ask you a favour" vs. "I would like to ask you for a favour"
Solution 1:
According to this Google NGram, all of the following forms are in use:
- ask you a favo(u)r
- ask you for a favo(u)r
As you can see, the forms without the preposition for are more common (with "ask you a favor" generating the highest results). In common use, one may conclude that this is the preferred form.
Solution 2:
BBC Learning English uses just the former expression for asking a favour.
Solution 3:
I would say the former is preferred/more common in colloquial/spoken English. The latter is not incorrect but used less often.