Here's a [direct link](http://www.math.rochester.edu/people/faculty/doug/otherpapers/axiomatic.pdf) 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:

1. 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](http://www.math.uni-bielefeld.de/~oroendig/MZfinal.pdf).

2. 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](http://arxiv.org/pdf/0909.1943.pdf).

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.