We propose a unified approach based on a bivariate linear mixed effects model to estimate three types of bivariate correlation coefficients (BCCs), as well as the associated variances between two quantitative variables in cross‐sectional data from a family‐type clustered design. These BCCs are defined at different levels of experimental units including clusters (e.g., families) and subjects within clusters and assess different aspects on the relationships between two variables. We study likelihood‐based inferences for these BCCs, and provide easy implementation using standard software SAS. Unlike several existing BCC estimators in the literature on clustered data, our approach can seamlessly handle two major analytic challenges arising from a family‐type clustered design: (1) many families may consist of only one single subject; (2) one of the paired measurements may be missing for some subjects. Hence, our approach maximizes the use of data from all subjects (even those missing one of the two variables to be correlated) from all families, regardless of family size. We also conduct extensive simulations to show that our estimators are superior to existing estimators in handling missing data or/and imbalanced family sizes and the proposed Wald test maintains good size and power for hypothesis testing. Finally, we analyze a real‐world Alzheimer's disease dataset from a family clustered study to investigate the BCCs across different modalities of disease markers including cognitive tests, cerebrospinal fluid biomarkers, and neuroimaging biomarkers.