With quilting, the fabric really doesn't have to be totally squared or "on grain". "On grain" matters more when sewing clothing, as grain affects the draping of garments. Being a little off-grain on the smallish pieces of fabric we use to piece quilts does not create a problem. You do not want your cuts to be on the bias (totally off-grain) because of the tendency for bias edges to stretch.
In your example above, I would simply try to get near-grain on one side or the other. If you have even selvedges and a flat fold, that is enough. At that point any strips you cut will be near-enough on-grain to not matter. What you need to be careful of when cutting strips is to make sure your ruler is at an accurate 90 degrees from the fold. If it isn't, you will get the dreaded "V" bend in your strip where the fabric was folded.