No they won't – in the frame of reference of the galaxy, both you and your pursuer will spend most of the journey traveling at very close to the speed of light. The difference won't be enough to catch up.
Put another way, in the frame of reference of the galaxy, you get to Andromeda (2.5 million light-years away) only about two years later than light would. Your pursuer, traveling at 1.1 g, would get there about a year and 9 months after light. So anyone chasing you would have to take off within about three months – and after a year, nothing could catch you.
This arrival time delay relative to light is pretty much constant (for fixed g) for any distance over about 1000 light years. For this reason, sci-fi authors who want ships to be overtaken by later models either slow the old ships down to much less than light-speed (i.e. generation ships) or introduce FTL travel.