In this paper, we prove the unique existence and investigate the $L^{p}$-regularity of solutions to stochastic partial differential equations in Hilbert spaces associated with pseudo-differential operators, driven by Hilbert space-valued Gaussian processes that satisfy certain regularity conditions for the covariance kernels of the Gaussian processes. For our purposes, we develop an $L^{p}$-regularity framework for the solutions to the stochastic partial differential equations associated with pseudo-differential operators. As the main tools, we establish the $p$-th moment maximal inequality for stochastic integrals with respect to a Hilbert space-valued Gaussian process and a Littlewood-Paley type inequality for Banach space-valued functions. Additionally, during our study, we improved the sufficient conditions for Fourier multipliers and examined the covariance kernels for Gaussian processes.