The emerging field of quantum imaging investigates the fundamental constraints imposed on imaging by quantum mechanics, and methods to overcome these constraints and thereby improve performance. In particular, non-classical states of light are known to allow both the shot-noise and diffraction limits to be surpassed; a prospect which is particularly attractive for biological imaging. However, quantum imaging technology is in its infancy and significant advances are still required for it to compete with its conventional counterparts. Although several recent experiments have demonstrated quantum enhanced imaging, these have improved image contrast in a regime many orders of magnitude worse than that achieved routinely in conventional imaging. We have developed an alternative technology which performs quantum enhanced particle tracking with high precision1. Using this, thermal vibrations of nanoparticles within living yeast cells were tracked, which allows the viscoelasticity of a small region of the cell to be precisely characterized. As the nanoparticle diffuses through an extended region of the cell, its motion can be used to map a spatial viscoelastic profile. Our experiment showed that subcellular structure as small as 10nm could be observed via its local influence on the thermal motion of nanoparticles2, which is comparable to the state-of-the-art. Furthermore, we demonstrated quantum enhanced resolution for the first time in biology, with non-classical light allowing 14% narrower resolution2. With improved technology, an order of magnitude enhancement in resolution may be possible. This demonstrates the potential for biological quantum imaging, though further advances are still required to make this technique practical.