One of the hallmarks of quantum mechanics is the impossibility of simultaneous measurement of non-commuting observables with projective measurements. This, however, can be circumvented by using continuous quantum measurements. Here we investigate the temporal correlations of the output signals of detectors continuously and simultaneously measuring the qubit observables σ_z and σ_z cosφ + σ_x sinφ, for various angles φ. Using the quantum Bayesian formalism, we obtain analytical expressions for the correlators, which we find to be in good agreement with those obtained from experimentally measured output signals. The agreement is particularly good for cross-correlators, even at times shorter than the cavity modes decay time. We further discuss how the correlators can be applied for parameter estimation, and use them to infer a small residual qubit Hamiltonian arising from calibration inaccuracy in the experimental data. Our work opens up new possibilities to perform quantum metrology based on temporal correlations of measured data.