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The Higgs mechanism, i.e., spontaneous symmetry breaking of the quantum vacuum, is a cross-disciplinary principle, universal for understanding dark energy, antimatter and quantum materials, from superconductivity to magnetism. Yet, Higgs modes in one-band superconductors (SCs) are currently under debate due to their competition with charge-density fluctuations. A distinct Higgs mode, controllable by terahertz (THz) laser pulses, can arise in multi-band, unconventional SCs via strong {\em interband} Coulomb interaction, but is yet to be accessed. Here we both discover and demonstrate quantum control of such collective mode in iron-based high-temperature superconductors. Using two-pulse, phase coherent THz spectroscopy, we observe a tunable and coherent 2$Δ_{\mathrm{SC}}$ amplitude oscillation of the complex order parameter in such SC with coupled lower and upper bands. The nonlinear dependence of the amplitude mode oscillations on the THz driving fields is distinct from any one-band and conventional SC results: we observe a large nonlinear change of resonance strength, yet with a persisting mode frequency. We argue that this result provides compelling evidence for a transient coupling between the electron and hole amplitude modes via strong interband coherent interaction. To support this scenario, we perform quantum kinetic modeling of a hybrid Higgs mechanism without invoking extra disorder or phonons. In addition to distinguishing between collective modes and charge fluctuations, the light quantum control of multiband SCs can be extended to probe and manipulate many-body entanglement and hidden symmetries in different quantum materials.