The integration of communication and radio-sensing functionalities on the same network has attracted attention recently, a paradigm designated as integrated sensing and communication (ISAC). This article addresses the problem of fully connected hybrid beamforming design for a multiuser and multibeam ISAC scenario. Previous methods of fully connected hybrid beamforming usually deal with this problem in two steps. First, the fully digital (FD) beamformer is obtained and then the hybrid beamformer is selected to minimize the distance to the FD counterpart. However, this approach may exhibit some drawbacks since the original communication and radio-sensing requirements may not be preserved. In contrast in this work, the fully connected hybrid beamforming is designed to ensure that communication and radio-sensing-related constraints are always fulfilled. The considered optimization criterion is the maximization of the weighted sum-rate subject to power budget and radio-sensing constraints. To address this problem, we propose a novel and convergent iterative alternate optimization algorithm to design the hybrid beamforming matrices that satisfy the criteria for both communication and radio sensing. The simulation results have shown that the performance is close to the FD precoder and outperforms other previous fully connected hybrid beamforming design methods for ISAC.