  We study the ground-state phase diagram of the strongly interacting
Harper-Hofstadter-Mott model at quarter flux on a quasi-one-dimensional lattice
consisting of a single magnetic flux quantum in $y$-direction. In addition to
superfluid phases with various density patterns, the ground-state phase diagram
features quasi-one-dimensional analogues of fractional quantum Hall phases at
fillings $\nu=1/2$ and $3/2$, where the latter is only found thanks to the
hopping anisotropy and the quasi-one-dimensional geometry. At integer fillings
- where in the full two-dimensional system the ground-state is expected to be
gapless - we observe gapped non-degenerate ground-states: At $\nu=1$ it shows
an odd 'fermionic' Hall conductance, while the Hall response at $\nu=2$
consists of the transverse transport of a single particle-hole pair, resulting
in a net zero Hall conductance. The results are obtained by exact
diagonalization and in the reciprocal mean-field approximation.
