The aim of this paper is to advance the numerical investigation of unsteady thermocapillary migration of gas bubbles in a fully confined non-isothermal liquid. The relevance of the study is twofold: to support space experiments such as the Italian experiment on board of the Spacelab IML-2 Mission and to fill a lack in thermocapillary migration studies mainly devoted to slow motion in infinite medium (so that no comparison is possible with experimental results). The motion due to surface tension gradients of one and two spherical bubbles inside a liquid filling a cylindrical box, with differentially heated ends disks, is herein analysed, both for linear and quadratic temperature dependence of the surface tension at the bubble-liquid interface. A Galerkin finite elements method is employed to solve the unsteady axisymmetric governing equations casted under a velocity, vorticity, temperature formulation. The flow field into the liquid medium, the net force exerted on the bubbles, the bubbles speed and location are determined as a function of the time; pressure and temperature fields around the bubble are reported.