The emergence of a collective behavior in a many-body system is responsible for the quantum criticality separating different phases of matter. Interacting spin systems in a magnetic field offer a tantalizing opportunity to test different approaches to study quantum phase transitions. In this work, we exploit the new resources offered by quantum algorithms to detect the quantum critical behavior of fully connected spin-1/2 models. We define a suitable Hamiltonian depending on an internal anisotropy parameter γ that allows us to examine three paradigmatic examples of spin models, whose lattice is a fully connected graph. We propose a method based on variational algorithms run on superconducting transmon qubits to detect the critical behavior for systems of finite size. We evaluate the energy gap between the first excited state and the ground state, the magnetization along the easy axis of the system, and the spin-spin correlations. We finally report a discussion about the feasibility of scaling such approach on a real quantum device for a system having a dimension such that classical simulations start requiring significant resources.