Is there a reference that lists/discusses examples of absolutely indecomposable representations of quivers over a finite field (absolutely indecomposable = does not decompose into a direct sum over the algebraic closure of the finite field, they are important in Kac's conjecture)?
Two references, neither of which exactly addresses your question, are as follows:
Ringel, Claus Michael Exceptional modules are tree modules. Linear Algebra Appl. 275/276 (1998), 471–493.
In this paper, Ringel shows that any exceptional representation can be written with a matrix using only 0's and 1's. In particular, it can be defined over a finite field. So this might be kind of the reverse of what you asked, but it does show that there is a nice class of representations over the algebraic closure which descend to representations over the finite field.
The textbook by Deng, Du, Parshall, and Wang, Finite dimensional algebras and quantum groups, develops a considerable amount of the theory of quiver representations over an arbitrary field, and includes material about the behaviour of representations over finite fields and how they split up over the algebraic closure. I don't remember seeing anything there specifically about absolutely indecomposable representations, but it should be possible to glean useful things.