In discrete models describing growing rough interfaces in the Kardar-Parisi-Zhang universality class, we examine height fluctuations at a fixed site as a function of time, measured in monolayer units. For small systems, we show that it is possible to reach the stationary state. We compute the two-time autocorrelation and power spectra independently. The correlation function remains non-exponential and vanishes after a correlation time that diverges with system size. As a result, the power spectra display a lower cutoff with constant power. In the nontrivial frequency regime, we observe (1/f^{})-type scaling with the spectral exponent 5/3. Finite-size scaling reveals that the temporal correlation function follows dynamic scaling. Our findings, supported by scaling-theoretical arguments, establish that the fluctuations are wide-sense stationary, implying the applicability of the Wiener-Khinchin theorem. The scaling arguments also reveal a scaling relation (= 1 + 2/z), where () and (z) denote the roughness and dynamical exponents.