Two Purely Inorganic Cationic Tellurite Materials Based on Group IB Metal Tetrafluoroborates with Similar Lamellar Cationic Layers: [Cu2F(Te2O5)](BF4) and [Ag18O2(Te4O9)4(Te3O8)(BF4)2]·2HBF4

Two new purely inorganic cationic tellurite networks of Group IB metal-based tetrafluoroborates, namely, [Cu2F(Te2O5)](BF4), 1, and [Ag18O2(Te4O9)4(Te3O8)(BF4)2]·2HBF4, 2, have been hydrothermally synthesized under mild conditions. The prepared materials have been characterized by single-crystal X-ray diffraction, powder X-ray diffraction, IR and Raman spectroscopy, SEM–energy-dispersive spectroscopy, UV–vis–NIR diffuse reflectance, magnetic study, and TG analyses. Single-crystal diffraction studies show that both materials have similar cationic Cu/Ag tellurite layers with tetrafluoroborates as interlamellar charge-balancing anions. Magnetic results indicate that [Cu2F(Te2O5)](BF4), 1, exhibits a mainly short-range antiferromagnetic ordering within the 2D layer, and further detailed analysis of magnetic susceptibility analysis confirms a spin-singlet ground state with an energy gap of 85 K.