We analyzed the triple Higgs boson self-coupling at future e ⁺ e ^- colliders energies, with the reactions e ⁺ e ^- → b \bar b H H , t \bar t H H . We evaluate the total cross-sections for both b \bar b H H and t \bar t H H , and calculate the total number of events considering the complete set of Feynman diagrams at tree-level. We vary the triple coupling κλ _3H within the range κ=-1 and +2. The numerical computation is done for the energies expected to be available at a possible Future Linear e ⁺ e ^- Collider with a center-of-mass energy 800, 1000, 1500 GeV and a luminosity 1000 fb ^-1 . Our analysis is also extended to a center-of-mass energy 3 TeV and luminosities of 1000 and 5000 fb ^-1 . We found that for the process e ⁺ e ^- → b \bar b H H , the complete calculation differs only by 3% from the approximate calculation e ⁺ e ^- → Z H H ( Z → b \bar b ), while for the process e ⁺ e ^- → t \bar t H H , the expected number of events, considering the decay products of both t and H , is not enough to obtain an accurate determination of the triple Higgs boson self-coupling.