It is known that the partial maximum of nonstationary Gaussian sequences converges in distribution and that the number of exceedances of a boundary is asymptotically a Poisson random variable, under certain restrictions. We investigate the rate of Poisson approximation for the number of exceedances. We generalize the result known in the stationary case, showing that the given bound of the rate depends on the largest positive auto-correlation value (less than 1) and the lowest values of the nonconstant boundary. We show that for special cases this bound cannot be improved.