If $\mathcal{C}$ is the category of commutative unitary monoids, one can endow the category of simplicial objects in $\mathcal{C}$, $s\mathcal{C}$, with the structure of a cofibrantly generated model category, in which fibrations are those maps inducing fibrations of the underlying simplicial sets (on the other hand, weak equivalences are possibly not precisely those maps inducing weak equivalences on underlying simplicial sets, as commutative monoids are not always Kan fibrant).

Is the homotopy category $\text{Ho} \mathcal({C})$ triangulated?

Perhaps it is pre-triangulated, at best, but the suspension functor is not necessarily an equivalence?

onlythe fibrations, and from your parenthetical it sounds like you aren't just lifting the model structure from simplicial sets (otherwise equivalences would be detected below, independent of fibrancy.) $\endgroup$ – Dylan Wilson Aug 26 '17 at 23:19