Since you're not asking how to level the floor, and since that would be a much more substantial project requiring much more information, I'll just address the molding detail.
Base shoe exists in part because it bends on the vertical plane much more readily than taller base trim does. It can be made to fit floors that aren't flat without planing or other cuts. Therefore, you just need to pull it off, push it down, and nail it back on. It'll usually flex to fit the general shape of your floor and look much better afterward. You may need to use longer nails and angle them into the subfloor.
In severe cases, where the shoe simply won't flex enough, you have three options:
- Ignore it. This is a lovely old home and it has character.
- Put a miter or two in the shoe to add slight angles. This may serve to draw the eye to the flaw, though.
- Caulk or fill it. Again, this may mostly draw attention to the flaw in the flooring. You'd want to use a tool such as a putty knife to get a nice flat surface. If you do so, it may make sense to paint the filled portion to match the flooring instead of leaving it white. You could also create a filler of natural wood and stain it to match the floor.
You might sand the base trim a bit before reinstallation to remove the paint edge that will remain, and you could give the shoe itself a bit of a sand to clean it up. Then install, caulk the joint on top if you like, and paint. Of course, new shoe molding would be an even better improvement since yours looks rather beat up. This wouldn't adversely affect the vintage look.
By the way, "quarter round" is a molding profile. "Base shoe" is quarter round (or other profiles) installed over base trim to conceal gaps in flooring and/or add detail. Since not all base shoe is quarter round, and since quarter round can be used for other purposes, base shoe is the better term here (or "shoe molding").