  In 2005, Nualart and Peccati showed the so-called Fourth Moment Theorem
asserting that, for a sequence of normalized multiple Wiener-It\^o integrals to
converge to the standard Gaussian law, it is necessary and sufficient that its
fourth moment tends to 3. A few years later, Kemp et al. extended this theorem
to a sequence of normalized multiple Wigner integrals, in the context of the
free Brownian motion. The q-Brownian motion, q in (-1,1], introduced by the
physicists Frisch and Bourret in 1970 and mathematically studied by Bozejko and
Speicher in 1991, interpolates between the classical Brownian motion (q=1) and
the free Brownian motion (q=0), and is one of the nicest examples of
non-commutative processes. The question we shall solve in this paper is the
following: what does the Fourth Moment Theorem become when dealing with a
q-Brownian motion?
