We study the ground state of the model Hamiltonian of the trimerized S = 1/2 quantum Heisenberg chain Cu₃Cl₆(H₂O)₂2H₈C₄SO₂ in which a non-magnetic ground state was observed recently. This model consists of stacked trimers and has three kinds of coupling constant describing the couplings between spins: the intra-trimer coupling constant J₁ and the inter-trimer coupling constants J₂ and J₃. All of these constants are assumed to be antiferromagnetic. By use of an analytical method and physical considerations, we show that there are three phases on the ₂ - ₃ plane (₂J₂ / J₁, ₃J₃ / J₁): the dimer phase, the spin-fluid phase and the ferrimagnetic phase. The dimer phase is caused by the frustration effect. In the dimer phase, there exists an excitation gap between the twofold-degenerate ground state and the first excited state, which explains the non-magnetic ground state observed in Cu₃Cl₆(H₂O)₂2H₈C₄SO₂. We also obtain the phase diagram on the ₂ - ₃ plane from the numerical diagonalization data for finite systems by use of the Lanczos algorithm.