I don't think this is actually fully answered by HA 1.4.4.2, which says that a stable $\infty$-category is presentable if and only if it's got coproducts (hence is cocomplete) and a $\kappa$-compact generator together with a locally small homotopy category (so that it's locally small.) That last condition sounds as if it would mean the homotopy category was $\kappa$-well generated, but there's no clear implication between being $\kappa$-compact in a stable $\infty$-category and being $\kappa$-compact in its underlying triangulated category. That's because being $\kappa$-compact in the homotopy category means being in the largest class of $\kappa$-small objects that is "$\kappa$-perfect" in the sense of Neeman, and it's only $\kappa$-smallness that has any clear connection to $\kappa$-filtered colimits.

However, a nearby claim is known, and was proved via a different model by Heider in his paper *Two results from Morita theory of stable module categories*. Heider shows that the homotopy category of a spectral model category is well generated if *and only if* it's triangulated equivalent to the localization of the derived category of a small spectral category at a set of objects. The derived category of a small spectral category models an arbitrary finitely presentable stable $\infty$-category, and its localizations at small sets model arbitrary accessible localizations, thus arbitrary $\alpha$-presentable stable $\infty$-categories.

Now, I'm pretty sure it's unknown precisely which stable $\infty$-categories underlie spectral model categories, so this doesn't quite answer the original question. I think it would be possible to imitate Heider's proof in general, though. Given an $\alpha$-compact generator $\mathcal G$ of the homotopy category of a stable $\infty$-category $\mathcal T$, seen as a full additive subcategory, take a small spectral category $\mathcal E$ with homotopy coherent nerve equivalent to the full subcategory of $\mathcal T$ spanned by the objects of $\mathcal G$. Then as far as I can tell, Heider's argument goes through without change to exhibit $\mathcal T$ as (the homotopy coherent nerve of) an $\alpha$-accessible localization of $D(\mathcal E^{\mathrm{op}})$.

Higher Algebra, Corollary 1.4.4.2. $\endgroup$ – Dylan Wilson Jan 4 '18 at 20:48