I would like to see results derived in hyperbolic trigonometry synthetically, i.e. just by working from axioms, for example the ones given by Hilbert (or even Tarski). Most authors seem to discuss hyperbolic trigonometry by working in a specific model, for example Hartshorne in Geometry: Euclid and Beyond gives a lot of results that model Hilbert's axioms but he derives hyperbolic trigonometry by considering the conformal disk model. I have read that Lobachevsky (and Minding) derived trigonometric identities by working from Euclid. Unfortunately I don't think their relevant papers have been translated to English so I wonder if some modern authors have tried this approach. Does somebody know of relevant paper or a textbook?
Have you looked at Lobachevsky's Geometrical researches on the theory of parallels (here a freely available translation by Halsted)? I hadn't so far, but skimming it now, it looks like a likely candidate for what you are asking, since it starts off in an axiomatic way, and does reach trigonometric formulas towards the end.
Formal proof assistant community
If you are looking for more recent works, then I'd consider possible motivations for choosing this approach. One such motivation that I can think of would be using proof assistants to work with these in a very formal way. Adding some likely keywords from that community, I found very recent papers like e.g. the simplest axiom system for plane hyperbolic geometry revisited, again by Alama from October 2013. Using something like this as an entry point, and following references back and forth (Google Scholar will be very useful there), you might also find one that does trigonometry in addition to axioms.
Another thing to look at might be the list of works citing the above mentioned Geometrical researches on the theory of parallels. There is a lot of math history work there, obviously, but also an Invitation to Elementary Hyperbolic Geometry by Zhang which has a section 4 on Hyperbolic Trigonometry. The introduction of that work states:
We choose this synthetic approach and as far as possible use no analytic models […]
Section 4 in fact does refer to the upper half plane model, but it does so in subsection 4.3, after deriving basic trigonometric identities in subsection 4.2.
That article also comes with a reading recommendation:
The 1968 edition of Hilbert's Grundlagen der Geometrie by Paul Bernays appendix 3 contains an article from the Mathematische Annalen, vol. 57, where Hilbert indeed founds the geometry of the Bolyai-Lobačevskij plane upon axioms. I find the text quite concise and therefore difficult, but -of course- tremendously interesting. I think that an English translation exists.