A mapping between fractional quantum Hall (FQH) junctions and the two-channel Kondo model is presented. We discuss in detail this relation for the particular case of a junction of a FQH state at $ν = 1/3$ and a normal metal. We show that in the strong coupling regime this junction has a non-Fermi liquid fixed point. At this fixed point the electron Green's function has a branch cut, and the entropy has a non-zero value equal to $S={1/2} \ln{2}$. We construct the space of perturbations at the strong coupling fixed point and find that the dimensions of the tunneling operator is 1/2. These behaviors are strongly reminiscent of the non-Fermi liquid fixed point of a number of quantum impurity models, particularly the two-channel Kondo model. However we have found that, in spite of these similarities, the Hilbert spaces of these two systems are quite different. In particular, although in a special limit the Hamiltonians of both models are the same, their Hilbert spaces are not since they are determined by physically distinct boundary conditions. As a consequence the spectrum of operators in both models is different. Comment: 12 pages, 4 eps figures. RevTex