We report on pressure-induced unconventional superconductivity (SC) in the heavy-fermion (HF) antiferromagnet CeIn3 by means of nuclear-quadrupole-resonance (NQR) studies conducted under a high pressure. The temperature (T) and pressure (P) dependences of the In-NQR spectra have revealed a first-order quantum-phase transition (QPT) from antiferromagnetism (AFM) to paramagnetism (PM) at a critical pressure P-c=2.46 GPa at which AFM disappears with a minimum value of T-N(P-c)=1.2 K. High-energy x-ray scattering measurements under P show a progressive decrease in the lattice density without any change in the crystal structure, whereas an increase in the NQR frequency (nu(Q)) indicates an increase in the hybridization between 4f electrons and conduction electrons, which stabilizes the HF-PM state. This competition between the AFM phase where T-N is reduced and the formation of the HF-PM phase triggers the first-order QPT at P-c=2.46 GPa. Despite the lack of an AFM quantum critical point in the P-T phase diagram, we highlight the fact that unconventional SC occurs in both phases of AFM and PM. The measurements of the nuclear spin-lattice relaxation rate 1/T-1 in the AFM phase have provided evidence for the uniformly coexisting AFM+SC phase. Remarkably, the significant increase in 1/T-1 upon cooling in the AFM phase has revealed the development of low-lying magnetic excitations down to T-c in the AFM phase; it is indeed relevant to the onset of the uniformly coexisting AFM+SC phase. In the HF-PM phase where AFM fluctuations are not developed, 1/T-1 decreases without the coherence peak just below T-c, followed by a power-law-like T dependence that indicates an unconventional SC with a line-node gap. Remarkably, T-c has a peak around P-c in the HF-PM phase as well as in the AFM phase. In other words, an SC dome exists with a maximum value of T-c=230 mK around P-c, indicating that the origin of the pressure-induced HF SC in CeIn3 is not relevant to AFM spin fluctuations but to the emergence of the first-order QPT in CeIn3. These phenomena observed in CeIn3 should be understood in terms of the first-order QPT because these new phases of matter are induced by applying P. When the AFM critical temperature is suppressed at the termination point of the first-order QPT, P-c=2.46 GPa, the diverging AFM spin-density fluctuations emerge at the critical point from AFM to PM. The results with CeIn3 leading to a new type of quantum criticality deserve further theoretical investigations.