  The generalized volume conjecture and the AJ conjecture (a.k.a. the quantum
volume conjecture) are extended to $U_q(\fraksl_2)$ colored quantum invariants
of the theta and tetrahedron graph. The $\SL(2,\bC)$ character variety of the
fundamental group of the complement of a trivalent graph with $E$ edges in
$S^3$ is a Lagrangian subvariety of the Hitchin moduli space over the Riemann
surface of genus $g=E/3+1$. For the theta and tetrahedron graph, we conjecture
that the configuration of the character variety is locally determined by large
color asymptotics of the quantum invariants of the trivalent graph in terms of
complex Fenchel-Nielsen coordinates. Moreover, the $q$-holonomic difference
equation of the quantum invariants provides the quantization of the character
variety.
