  A construction method of infinite families of quasi-exactly solvable
position-dependent mass Schr\"odinger equations with known ground and first
excited states is proposed in a deformed supersymmetric background. Such
families correspond to extensions of known potentials endowed with a deformed
shape invariance property. Two different approaches are combined. The first one
is a generating function method, which enables to construct the first two
superpotentials of a deformed supersymmetric hierarchy, as well as the first
two partner potentials and the first two eigenstates of the first potential,
from some generating function $W_+(x)$ [and its accompanying function
$W_-(x)$]. The second approach is the conditionally deformed shape invariance
method, wherein the deformed shape invariance property of the starting
potentials is generalized to their extensions by adding some constraints on the
parameters and by imposing compatibility conditions between sets of
constraints. Detailed results are given for some extensions of the linear and
radial harmonic oscillators, as well as the Kepler-Coulomb and Morse
potentials.
