  In this paper, we construct the phase space of a constantly curved
tetrahedron with fixed triangle areas in terms of a pair of Darboux coordinates
called the length and twist coordinates, which are in analogy to the
Fenchel-Nielsen coordinates for flat connections, and their quantization. The
curvature is identified to the value of the cosmological constant, either
positive or negative. The physical Hilbert space is given by the
$\mathcal{U}_q(\mathfrak{su}(2))$ intertwiner space. We show that the quantum
trace of quantum monodromies, defining the quantum length operators, form a
fusion algebra and describe their representation theory. We also construct the
coherent states in the physical Hilbert space labeled by the length and twist
coordinates. These coherent states describe quantum curved tetrahedra and peak
at points of the tetrahedron phase space. This works is closely related to 3+1
dimensional Loop Quantum Gravity with a non-vanishing cosmological constant.
The coherent states constructed herein serve as good candidates for the
application to the spinfoam model with a cosmological constant.
