Here's a direct link to the book by Hovey–Palmieri–Strickland.

The category of motivic spectra is known to satisfy the axioms of Definition 1.1.4 in the book when the base is a countable field of characteristic zero. Axioms (c) and (e) are problematic in general:

Motivic spectra are generated by strongly dualizable objects when the base is a field of characteristic zero. The proof uses Hironaka's resolutions of singularities, see Röndigs–Østvær, Modules over motivic cohomology.

The representability of cohomology functors holds if the base is covered by finitely many spectra of countable commutative rings. See Naumann–Spitzweck, Brown representability in A1-homotopy theory.

My 2 cents: ~~axiom (c) sounds like a reasonable conjecture over general base schemes~~, but it seems very unlikely that axiom (e) would hold beyond the known case.

**ETA** In case someone stumbles upon this answer, let me take back one cent and say that axiom (c) should not hold over any scheme of positive dimension. The correct expectation is that it holds over zero-dimensional schemes (this being only known in characteristic zero for now). The idea is that the subcategory generated by dualizable object is the category of *locally constant* motivic spectra. Here is a proof that (c) does not hold for $SH(S)$ assuming $S$ admits a regular codimension 1 point $s$ (eg $S$ is normal): take an open $U$ whose restriction to $\mathrm{Spec}(\mathcal{O}_{S,s})$ is the generic point. Then $\Sigma^\infty_T U_+\in SH(S)$ is not generated by dualizable objects; in fact its rational motive isn't: localization and absolute purity imply easily that pullback to the generic point $DM_{\mathbb Q}(\mathcal{O}_{S,s})\to DM_{\mathbb Q}(\eta)$ is conservative on locally constant motivic sheaves.