  It is considered the integrated process $X(t)= x + \int _0^t Y(s) ds ,$ where
$Y(t)$ is a Gauss-Markov process starting from $y.$ The first-passage time
(FPT) of $X$ through a constant boundary and the first-exit time of $X$ from an
interval $(a,b)$ are investigated, generalizing some results on FPT of
integrated Brownian motion. An essential role is played by a useful
representation of $X,$ in terms of Brownian motion which allows to reduces the
FPT of $X$ to that of a time-changed Brownian motion. Some explicit examples
are reported; when theoretical calculation is not available, the quantities of
interest are estimated by numerical computation.
