(covers information from several alternate timelines)
Greer was a descendant of the crew of Enterprise NX-01 in an alternate timeline in which Enterprise was thrown back in time from the year 2154 to 2037. As of 2154, Greer was serving as Enterprise's tactical officer, under the command of Lorian, the son of T'Pol and Commander Charles Tucker.
In that year, the crew intercepted their counterparts in an effort to prevent the accident which had thrown them back in time in the first place. Greer participated in Lorian's attempt to steal their counterparts' modified plasma injectors, which they planned to install on their own ship in order to make a rendezvous with the Xindi-Primate Degra. Greer manned the tactical station during the subsequent battle with their counterparts, but when it became evident that Captain Archer and his crew had gained the upper hand, Lorian surrendered. Greer later participated in both ships' battle against the Kovaalans, an aggressive species which controlled a nebula in which a subspace corridor was located. Lorian and his crew successfully held off the Kovaalans as Archer's Enterprise entered the corridor. Archer and his crew waited for five hours for the other Enterprise to emerge from the corridor, before it became clear that they would not be arriving. The fate of Greer and the other Enterprise remained unknown. Although it seemed likely that they were destroyed by the Kovaalans, Archer also entertained the idea that their existence was negated due to the fact that the initial accident which transported Enterprise back in time in the first place had been averted. (ENT: "E²")