We point out that a high number density of stars in the core of a dense star cluster such as the central stellar cluster at the Galactic center, where many stars possess strong stellar winds, should result in collisions of those winds. The wind collisions in the dense stellar core at the Galactic center would result in production of strong X-ray flares with the rate of ~ 10^-4 (N_w /10^3)^2 yr^-1 and duration of ~ 1 week, where N_w is the number of the wind producing stars in the core. Presence of a massive black hole would enhance the stellar density around it and would make collisions of the winds in the core substantially more frequent. Collisions of the stellar winds in the cluster have also a number of interesting observable implications, including generation of gamma-rays by particles accelerated by the shocks from the colliding winds. These processes are also expected to be relevant to compact regions of intense star formation elsewhere.