Why an all-Q-turret warship is a dead end

Would a battleship or cruiser armed only with Q turrets be viable?

No; historical central-battery layouts demonstrate how badly such an arrangement restricts forward and aft fire.

Historical central-battery arrangements show the problem with placing every main turret amidships. Their guns could not fire effectively ahead or astern. A fighting ship would lose the flexible arcs needed to engage targets across changing bearings. The layout may look orderly, but its restricted fire makes it a poor operational design.