A new sorption material (GS) was obtained by the modification of heulandite zeolite (G) with N,N′-bis-(3-triethoxysilylpropyl)thiocarbamide (S). The composition, structure, and surface morphology of the GS material were confirmed using elemental analysis, IR-, NMR-spectroscopy, X-ray diffraction, scanning electron microscopy (SEM), energy dispersive X-ray spectroscopy (EDX), elemental mapping, and nitrogen adsorption/desorption (BET). The potential of GS as a sorbent for the removal of Cu(II) and Ni(II) ions from concentrated solutions was demonstrated. The nature of the adsorption of Cu(II) and Ni(II) ions was investigated using the Langmuir, Freundlich, and Dubinin–Radushkevich models. The adsorption value of Cu(II) and Ni(II) ions by the GS sorbent was found to be 1.7 and 2.1 times higher than that of heulandite, amounting to 0.128 mmol/g (8.1 mg/g) and 0.214 mmol/g (12.6 mg/g), respectively. The free energy of adsorption E for the adsorption of Cu(II) and Ni(II) ions was determined to be 12.5 and 16.2 kJ/mol, respectively. Calculations of changes in Gibbs energy based on quantum chemical modeling results ($\Delta G_{298}^0$ = −38.5 kJ/mol for Ni and $\Delta G_{298}^0$ = −56.5 kJ/mol for Cu) confirmed that adsorption of heavy metal ions onto the GS sample occurs through the formation of metal ion coordination complexes with the sorbent’s functional groups (chemosorption). The proposed method of obtaining new sorption materials based on natural heulandite is straightforward and cost-effective, enabling the production of high-capacity sorption products.