The term "flush with" or "flush against" means "level with" or "even with," in the sense that an object is perfectly aligned with something else (usually another surface that it is touching). If the back of a chair is "flush against the wall," then it stands vertically at the same angle as the wall, so that there is little or no space between the back of the chair and the wall surface. The word is normally used for flat surfaces.
In this context, the author seems to be trying to show how firmly Morrie's head is pressed against the pillow, leaving no space in between.
(Edited to add: Adam explained it well, I just didn't see his answer until after I'd already sent mine!)