  The phase space representation for a q-deformed model of the quantum harmonic
oscillator is constructed. We have found explicit expressions for both the
Wigner and Husimi distribution functions for the stationary states of the
$q$-oscillator model under consideration. The Wigner function is expressed as a
basic hypergeometric series, related to the Al-Salam-Chihara polynomials. It is
shown that, in the limit case $h \to 0$ ($q \to 1$), both the Wigner and Husimi
distribution functions reduce correctly to their well-known non-relativistic
analogues. Surprisingly, examination of both distribution functions in the
q-deformed model shows that, when $q \ll 1$, their behaviour in the phase space
is similar to the ground state of the ordinary quantum oscillator, but with a
displacement towards negative values of the momentum. We have also computed the
mean values of the position and momentum using the Wigner function. Unlike the
ordinary case, the mean value of the momentum is not zero and it depends on $q$
and $n$. The ground-state like behaviour of the distribution functions for
excited states in the q-deformed model opens quite new perspectives for further
experimental measurements of quantum systems in the phase space.
