  Triply degenerate points (TDPs), which correspond to new types of topological
semimetals, can support novel quasiparticles possessing effective integer spins
while preserving Fermi statistics. Here by mapping the momentum space to the
parameter space of a three-level system in a trapped ion, we experimentally
explore the transitions between different types of TDPs driven by
spin-tensor--momentum couplings. We observe the phase transitions between TDPs
with different topological charges by measuring the Berry flux on a loop
surrounding the gap-closing lines, and the jump of the Berry flux gives the
jump of the topological charge (up to a $2\pi$ factor) across the transitions.
For the Berry flux measurement, we employ a new method by examining the
geometric rotations of both spin vectors and tensors, which lead to a
generalized solid angle equal to the Berry flux. The controllability of
multi-level ion offers a versatile platform to study high-spin physics and our
work paves the way to explore novel topological phenomena therein.
