Do Owlbears hibernate like bears or stay active all winter like owls? Even the full page description of Owlbears in the AD&D 2nd Edition Monstrous Manual is silent on this topic. I don't have back issues of Dragon Magazine available so I'm unsure if hibernation was ever discussed. The campaign I run is set in the Dalelands in the 1350s DR, a temperate climate for wild Owlbears, with plentiful caves, and ruins.
Yes, they hibernate
There are two sources for this, one is the 2nd Edition sourcebook Monstrous Manual, which has this on the ecology of the owlbear:
Ecology: Owlbears have a lifespan of 20 years. They are warm-blooded mammals, but lay eggs. They prey on anything, from rabbits to bears, to trolls, to snakes and reptiles. Owlbears prefer temperate climates, but some thrive in subarctic environments. As a hybrid of two animals, one diurnal and the other nocturnal, they have an unusual active time, waking at noon, hunting animals active during the day, then hunting nocturnal creatures before going to sleep at midnight. Owlbears are active in the summer months and hibernate during the cold season. There are rumors of white arctic owlbears, a cross between arctic owls and polar bears, but no specimens have ever been captured.
This is also somewhat corroborated by a (non official) tweet from Chris Perkins:
Whatever works for your world. Personally, I dig that they hunt at night like owls and hibernate like bears.
So, essentially the last part of what he said. You can go with official past lore and have them hibernate, or not hibernate, or any variant in between. If you do decide they hibernate, there are past publications to back up that stance.
Bears do not hibernate
I'm not aware of any D&D publication with a canonical answer, but modern science suggest that bears don't actually hibernate . While they may be less active, and "den up" during the winter they are still frequently active, and wake up easily. In some cases (especially Black Bears) they may enter a torpor like state. While this is fairly distinct from the kind of hibernation exhibited by most smaller mammals, only so much can be inferred about how a magical monstrosity will behave based on the creatures combined to create it. To some extent whether this is "hibernation" or not would be a matter of semantics.
It's your game, do what makes sense for your Narrative
So on the one hand it's an easy assumption that Owlbears are active (or at least semi-active) year round. However the flip side, being monstrous creatures of uncertain origin, you have fairly free reign as DM to decide this sort of information about their biology. Based on their description it's not unreasonable to have them hibernate for a month after every large kill. Or for the week after Selune is full.