In this paper a new Green's function and a new Poisson-type integral formula for a boundary value problem (BVP) in thermoelasticity for a quarter-space with mixed homogeneous mechanical boundary conditions are derived. The thermoelastic displacements are generated by a heat source, applied in the inner points of the quarter-space and by heat flux, prescribed on its boundary half-planes. All results are obtained in closed forms that are formulated in a special theorem. A closed form solution for a particular BVP of thermoelasticity for a quarter-space also is included. The main difficulties to obtain these results are in deriving the functions of influence of a unit concentrated force onto elastic volume dilatation (k) and, also, in calculating a volume integral of the product of function (k) and Green's function in heat conduction. Using the proposed approach it is possible to extend the obtained results not only for any canonical Cartesian domain, but also for any orthogonal one.