  Quantum discord, as a measure of all quantum correlations, has been proposed
as the key resource in certain quantum communication tasks and quantum
computational models without containing much entanglement. Daki\'{c}, Vedral,
and Brukner [Phys. Rev. Lett. 105, 190502 (2010)] introduced a geometric
measure of quantum discord (GMQD) and derived an explicit formula for any
two-qubit state. Luo and Fu [Phys. Rev. A 82, 034302 (2010)] introduced another
form of GMQD and derived an explicit formula for arbitrary state in a bipartite
quantum system. However, the explicit analytical expression for any bipartite
system was not given. In this work, we give out the explicit analytical
expressions of the GMQD for a two-parameter class of states in a qubit-qutrit
system and study its dynamics for the states under various dissipative channels
in the first time. Our results show that all these dynamic evolutions do not
lead to a sudden vanishing of GMQD. Quantum correlations vanish at an
asymptotic time for local or multi-local dephasing, phase-flip, and
depolarizing noise channels. However, it does not disappear even though t --->
$\infty$ for local trit-flip and local trit-phase-flip channels. Our results
maybe provide some important information for the application of GMQD in hybrid
qubit-qutrit systems in quantum information.
