Depends on the sort of owlbear
Most of the information on possible owlbear rearing in this answer is from The Ecology of the Owlbear in Dragon #214. The other information is from the Monstrous Manual and the Player's Handbook.
First of all, you may not have realized how smart owlbears are. Owlbears aren't just magical beasts, they are sentient magical beasts with their own language and quite intelligent. Specifically, they are of intelligence 'Low (5-7)' for all varieties, which is more intelligent than most animals and on par with your average troll, and more intelligent than what the Animal Training skill allows:
A character can choose even more exotic creatures and monsters with animal intelligence (although these are difficult to control).
(emphasis added)
Second of all, you should be aware that people have been actively trying to domesticate owlbears since their creation and they've yet to succeed even once, at least in the case of the standard and winged owlbears:
Owlbear eggs are worth 2,000 silver pieces and hatchlings are worth 5,000
silver pieces on the open market. These are typically bought by wizards; while they are impossible to
domesticate, they make formidable guardians and wizards sometimes place them in locations of strategic
importance (it has been said that an owlbear is a less subtle version of a "keep out" sign).
and
Winged owlbears tend to live slightly longer than the normal variety, often reaching 25 years or so. They are sought after by wizards, even though to date no one has managed to domesticate one; nonetheless, eggs can go as high as 4,000 silver pieces, while the price for live young can reach 10,000 silver pieces.
So, then, your best hope would be to have acquired the egg of the potentially gregarious Arctic Owlbear (while the Monstrous Manual lists the standard owlbear as a pack animal in the table as well, that's clearly out of touch with the detailed description in the text. Not so for the Artic Owlbear, which says nothing about forming mated pairs beyond the contents of such a pair's lair), but:
There are rumors of white arctic owlbears, a cross between
arctic owls and polar bears, but no specimens have ever been captured
Regardless of which creature you wish to futilely try to enslave, the appropriate non-weapon proficiency is probably still Animal Training (Owlbear), with the aforementioned 'you can't actually take this skill according to the rules' caveat. If I were GMing, though, I would instead require you to possess both Animal Training (Bear) and Animal Training (Owl), both of which are legal (though Bears are too intelligent to be normally affected by the skill, their statblock indicates they can nonetheless be trained).
Regarding the 'DC', that's not how non-weapon proficiencies work. A significant negative modifier to your ability score for the proficiency check(s) would be appropriate.
