  We study universal clusters in quasi-two dimensions (q2D) that consist of a
light (L) atom interacting with two or three heavy (H) identical fermions,
forming the trimer or tetramer bound state. The axial confinement in q2D is
shown to lift the three-fold degeneracy of 3D trimer (tetramer) in $p$-wave
channel and uniquely select the ground state with magnetic angular momentum
$|m|=1$ ($m=0$). By varying the interaction or confinement strength, we explore
the dimensional crossover of these clusters from 3D to 2D, characterized by a
gradual change of critical H-L mass ratio for their emergence and
momentum-space distribution. Importantly, we find that a finite effective range
will {\it not} alter their critical mass ratios in the weak coupling regime.
There, we establish an effective 2D model to quantitatively reproduce the
properties of q2D clusters, and further identify the optimal interaction
strengths for their detections in experiments. Our results suggest a promising
prospect for observing universal clusters and associated high-order correlation
effects in realistic q2D ultracold Fermi mixtures.
