  We investigate the quantum origin of the primordial cosmological
gravitational waves in the Brans-Dicke theory. We compute the number of
gravitons $N_k$ produced during inflation and the observables: power spectrum
$P_T$, spectral index $n_T$ and energy density $\Omega_k$. By comparison with
General Relativity we see that the results for both theories are the same for
the case of the particles number $N_k$ and for the case of the power spectrum
$P_T$ and the energy density $\Omega$ only when $\omega<10$. For the spectral
index $n_T$ we found that when the Brans-Dicke coupling parameter $\omega$ is
bigger than unity the spectral index approximates the expression found in the
General Relativity. This may awake us to the possibility that $\omega$ varies
with the cosmological scale.
