He must've immigrated to a planet there after he invented the warp drive.The real reason is because Metamorphosis came out in the 60s, and First Contact came out in the 90s. An in-universe explanation could be that Cochrane moved to Alpha Centauri later in life.
That is of course a common misconception. Kirk actually said, "OF Alpha Centauri," which is not necesarily the same thing as "From." For example, Lawrence OF Arabia was not actually FROM Arabia.