On this FAQ, I found that Diamond Moderators have ability to refund back bounties. Can they use this ability on their own bounties?
Yes, we can.
Now whether or not we should or do is a separate story. It would be wrong of us to reward our bounties back to ourselves as that means we are using our moderation abilities to do things that unfairly benefit us as users of the site.
Users (moderators or not) have two options if they want to ask for a bounty refund:
Regardless of the mechanism, if the user making the request is a moderator, then that moderator should (and by should, I mean is very strongly advised to) let a different moderator handle the request.