If the start time is the problem, and if the whole performance is off by the same amount, then it is not quantizing you need, but just shifting the performance to the right start time.
If that's it, the root cause could be that there is some inherent latency in the playback of whatever you are playing against in the project, be that a metronome or a prearranged accompaniment. Usually, a recording is aligned (after the fact) when played against a project with latency, but sometimes things can go wrong with that.