Which is correct? "Confident in" or "confident of"?

They are not interchangeable, strictly speaking. Loosely used, or in specific defining contexts, the senses do overlap.

Confident, adj. and n., from the OED

2. a. Having strong belief, firm trust, or sure expectation; feeling certain, fully assured, sure.
   ...
   c. Const. of (formerly with inf.).
   ...
   d. Const. in. (This has affinities with 3.)
   ...
3. Full of assurance, self-reliant, bold; sure of oneself, one's cause, etc.; having no fear of failure.

The meaning in isolation (without context) of 'confident of' conveys the sense of 'belief in', so your example

I am confident of my abilities

to me conveys that you believe in your abilities, but not necessarily that you act as if you believe in your abilities.

So, in contrast, your example

I am confident in my abilities

conveys more of a sense that when those abilities are in play, you act confidently.

The distinction I've drawn between the two echoes what the OED defines, perhaps more lucidly, by parenthetically mentioning the affinities of 2d., the construction with "in", with sense 3.