Discrete Tomography fits in the broad framework of inverse problems. One of its main issues concerns the faithful reconstruction of discrete homogeneous objects from internal quantitative information known as projections. In this paper, we consider specific types of projections that count, for each position, the number of points of the discrete objects that lie in its 8 or 4 neighbourhood, including the point itself. Our final goal is to reconstruct the object from the projections, if possible uniquely, or to determine the minimal number of its points that have to be revealed to do that. This problem is commonly addressed as the Minimum Surgical Probing problem (MSP). Our contribution is to study MSP with the addition of convexity constraint. In particular, we focus on a specific class of objects, commonly involved in Discrete Tomography, namely hv-convex polyominoes, which are connected sets of points with the further constraint of horizontal and vertical convexity. These constraints allow us to achieve unique reconstruction (except in some corner cases). Additionally, they allow for faster reconstruction algorithms.