I have a concrete question for the algebraic category of spectra, but if there is an answer for its topological analogue I would be interested in it.

Let $S$ be a finite dimensional noetherian scheme and $\mathbf{Spt}(S)$ the category of spectra over $S$. After inverting $\mathbb{A}^1$-stable equivalences we obtain Voevodsky's stable homotopy category $\mathbf{SH}(S)$. My question is:

Is there a model structure on $\mathbf{Spt}(S)$, having $\mathbf{SH}(S)$ as homotopy category, such that every object is fibrant? If so, could you provide a reference?

For example, the obvious candidate given by the class of $\mathbb{A}^1$-stable equivalences as weak equivalences, surjective morphisms as fibrations, and those having the adequate lifting property, defines a model structure on $\mathbf{Spt}(S)$?