  We consider an impurity problem in a quasi-two-dimensional Fermi gas, where a
spin-down impurity is immersed in a Fermi sea of N spin-up atoms. Using a
variational approach and an effective two-channel model, we obtain the energies
of the system for a wide range of interaction strength and for various
different mass ratios between the impurity and the back ground fermion in the
context of heteronuclear mixture. It is demonstrated that in a
quasi-two-dimensional Fermi gas there exists a transition of the ground state
from polaron in the weakly interacting region to molecule in the strongly
interacting region. The critical interaction strength of the polaron-molecule
transition is non-universal and depends on the particle density of the
background Fermi sea. We also investigate the properties of the excited
repulsive polaron state, and find similar non-universal behavior.
