The use of galaxy clusters as precision cosmological probes relies on an accurate determination of their masses. However, inferring the relationship between cluster mass and observables from direct observations is difficult and prone to sample selection biases. In this work, we use weak lensing as the best possible proxy for cluster mass to calibrate the Sunyaev-Zeldovich (SZ) effect measurements from the APEX-SZ experiment. For a well-defined (ROSAT) X-ray complete cluster sample, we calibrate the integrated Comptonization parameter, $Y_{rm SZ}$, to the weak-lensing derived total cluster mass, $M_{500}$. We employ a novel Bayesian approach to account for the selection effects by jointly fitting both the SZ Comptonization, $Y_{rm SZ}text{--}M_{500}$, and the X-ray luminosity, $L_{rm x}text{--}M_{500}$, scaling relations. We also account for a possible correlation between the intrinsic (log-normal) scatter of $L_{rm x}$ and $Y_{rm SZ}$ at fixed mass. We find the corresponding correlation coefficient to be $r= 0.47_{-0.35}^{+0.24}$, and at the current precision level our constraints on the scaling relations are consistent with previous works. For our APEX-SZ sample, we find that ignoring the covariance between the SZ and X-ray observables biases the normalization of the $Y_{rm SZ}text{--}M_{500}$ scaling high by $1text{--}2sigma$ and the slope low by $sim 1sigma$, even when the SZ effect plays no role in the sample selection. We conclude that for higher-precision data and larger cluster samples, as anticipated from on-going and near-future cluster cosmology experiments, similar biases (due to intrinsic covariances of cluster observables) in the scaling relations will dominate the cosmological error budget if not accounted for correctly.
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