I have existing select red oak hardwoods in foyer and dining room and am planning on installing unfinished select red oak in the greatroom, stripping the old floor and finishing both in the same finish. There will be a transition piece between the foyer and greatroom to handle the increase in height in the greatroom, so the old floor will not tie directly into the new floor.

From what I've read, quartersawn wood is more moisture stable since it changes more in thickness than in width.The existing floor is plainsawn. Would it look odd to use quartersawn red oak in the greatroom while keeping the plainsawn in the adjacent foyer and dining room?

IMO, yes. Q-sawn takes stain and finishes differently, even when using the same products. To make all the floors appear the same, if that is your goal, stick with the same type of wood and sawn style. Case in point: I recently did a job where I needed to add on to an existing red oak floor, plain sawn. I didn't know the brand but only that it was red oak 2&1/4" strip plain sawn. So when I was ordering the new material, the distributor suggested I use their "higher quality" and more expensive brand. So I ordered that. It turned out to be N. red oak instead of S. red oak and the color and graining was different. This wasn't noticeable until I stained the floors. Then it was too late. Fortunately, the customer was ok with this and liked the look of the N. red oak. This could have been a major problem and I would have had to deal with it. But even though the new floors look a little different than the older floors, the clients were happy and satisfied. And that is what matters.

Ok, thanks. I don't think my wife would be too happy if there is a significant difference in the appearance of the new vs. the old hardwood. The existing wood floor was put in 14 years ago and the only information I have on it is that it was supposed to be select red oak. Is there anyway to identify what kind of red oak (northern vs. southern vs. ? ) to avoid significant differences in stained appearance?

One suggestion would be to contact NOFMA or a NOFMA member for help.
www.nofma.org

They would be the ones to give you the help you need. Identification of Species is not a skill many have.

Good Luck