Quantifying uncertainty in measured properties of nanomaterials is a prerequisite for the manufacture of reliable nanoengineered materials and products. Yet, rigorous uncertainty quantification (UQ) is rarely applied for material property measurements with the atomic force microscope (AFM), a widely used instrument that can measure properties at nanometer scale resolution of both inorganic and biological surfaces and nanomaterials. We present a framework to ascribe uncertainty to local nanomechanical properties of any nanoparticle or surface measured with the AFM by taking into account the main uncertainty sources inherent in such measurements. We demonstrate the framework by quantifying uncertainty in AFM-based measurements of the transverse elastic modulus of cellulose nanocrystals (CNCs), an abundant, plant-derived nanomaterial whose mechanical properties are comparable to Kevlar fibers. For a single, isolated CNC the transverse elastic modulus was found to have a mean of 8.1 GPa and a 95% confidence interval of 2.7-20 GPa. A key result is that multiple replicates of force-distance curves do not sample the important sources of uncertainty, which are systematic in nature. The dominant source of uncertainty is the nondimensional photodiode sensitivity calibration rather than the cantilever stiffness or Z-piezo calibrations. The results underscore the great need for, and open a path towards, quantifying and minimizing uncertainty in AFM-based material property measurements of nanoparticles, nanostructured surfaces, thin films, polymers and biomaterials.