ALMA observations of the Galactic center with a spatial resolution of 2.″61 × 0.″97 resulted in the detection of 11 SiO (5-4) clumps of molecular gas within 0.6 pc (15″) of Sgr A*, interior to the 2 pc circumnuclear molecular ring. The three SiO (5-4) clumps closest to Sgr A* show the largest central velocities, ≃150 km s-1, and the broadest asymmetric line widths with full width zero intensity (FWZI) ≃110-147 km s-1. The remaining clumps, distributed mainly to the NE of the ionized mini-spiral, have narrow FWZI (≃18-56 km s -1). Using CARMA SiO (2-1) data, Large Velocity Gradient modeling of the SiO line ratios for the broad velocity clumps constrains the column density N(SiO) ≃1014 cm-2, and the H2 gas density cm -3 for an assumed kinetic temperature 100-200 K. The SiO clumps are interpreted as highly embedded protostellar outflows, signifying an early stage of massive star formation near Sgr A* in the last 104-10 5 yr. Support for this interpretation is provided by the SiO (5-4) line luminosities and velocity widths which lie in the range measured for protostellar outflows in star-forming regions in the Galaxy. Furthermore, spectral energy distribution modeling of stellar sources shows two young stellar object candidates near SiO clumps, supporting in situ star formation near Sgr A*. We discuss the nature of star formation where the gravitational potential of the black hole dominates. In particular, we suggest that external radiative pressure exerted on self-shielded molecular clouds enhances the gas density, before the gas cloud becomes gravitationally unstable near Sgr A*. Alternatively, collisions between clumps in the ring may trigger gravitational collapse.