Yes, it does so automatically. There will be an altimeter setting readout at the controller's position, and that's the setting that controller's display is using, and the one the controller will read to all aircraft in that sector. Note that this varies from sector to sector, so each controller may read you a different number. Further, with a big enough sector, there may be different settings within that same sector, so the controller may read you one setting while reading another to someone a distance away.
Most of the time the Denver Center altimeter setting is not the same (and more than 0.1" different) than KFTG. Distance between the two is about 35 nm.