In this paper, contact boundary condition between two pieces of the processed meat at different temperatures (two pieces with cold and room temperatures), that were suddenly in contact with each other in an inverse non-Fourier heat conduction problem was estimated. The direct problem was non-Fourier heat conduction that expressed with fractional single phase model and this problem involved only the piece of the processed meat in the room temperature. In this problem, it was assumed that governing equation, thermophysical properties and initial and boundary conditions were known and then it was solved numerically using the modified Mac-Cormack method. In the inverse problem, the estimation of the unknown contact boundary condition in the fractional non-Fourier model as a new work is done using the parameter estimation version of conjugate gradient method without/with adjoin problem that is one of the efficient deterministic methods in inverse analysis. In addition, in order to obtain the measured temperature of the inverse problem, a linear dual phase lag model validated with experimental data, was used. Finally, these two methods were compared to each other. Their results of these two methods showed the efficient estimation of the unknown contact boundary condition in fractional non-Fourier heat conduction.