Consider the category $Cat(S)$ of internal categories in a finitely complete category $S$ equipped with a Grothendieck pretopology $J$. For $S$ and $J$ satisfying certain properties, then there is a Quillen model structure, shown by [Everaert, Kieboom and van der Linden](http://www.tac.mta.ca/tac/volumes/15/3/15-03abs.html) where the weak equivalences are those internally fully faithful, essentially 'surjective' (*) functors. But without the asumptions that EKvdL give, there is only a Brown model structure in general (I don't know if this is published, Urs Schreiber mentioned it to me and I wrote down the proof myself). Mostly the problem comes with not having cocompleteness of $Cat(S)$, for which they give sufficient assumptions, like $S$ being a topos with NNO, or being a finitely cocomplete regular Mal'cev category. 

One might claim (Urs might, for example :) that this is a special case of a category of simplicial sheaves from Otgonbayar Uuye's answer, namely those simplicial sheaves that are nerves of categories and representable. But one does not need the machinery of simplicial sheaves to talk about this case.