If I focus on the center of a flat subject, will the edges be out of focus?

Ideally, a lens forms focus on a flat plane, so a flat subject that is parallel to the sensor should be in focus across the frame when you focus correctly.

In practice, lenses are not always perfect. A common aberration called field curvature means the plane of best focus can be slightly curved rather than flat. So depending on the lens, off-center areas may be a little less sharp even if the center is focused. Some lenses—especially many macro lenses and perspective-control lenses—are designed for a flatter field, while others may allow more curvature.

Also, focus is never truly “perfect” at only one exact distance in real-world use. There is always an acceptable range of sharpness, and stopping down to a smaller aperture increases depth of field and usually improves edge sharpness from slight focus errors or field curvature.

So yes, it can matter, especially at close distances, but whether it is significant depends on the lens design and aperture.