A ship with a functional Quantum Reactor can NEVER run out of supplies. So, having a supply storage component on it is useless. This is the question that was asked. Now for your question. You should always have at least one supply storage _OR_ a QR on all warships, to help against Ionic Dispersers. QRs share supplies sometime between game turns, so they do not help other ships in the fleet in combat. But having a QR on each ship means it will never be able to run out of supplies. So if all ships have a QR, you don't need to worry about it.
But if you have one ship in the fleet with a QR, then there is no chance of your ships running out of supplies in any combat unless their engines and all supply storage components are destroyed, in which case a QR can't ever help them.
All the ships start each turn fresh with full supplies, and no amount of combat and movement possible in a single turn can possibly deplete every Last supply from a ship that starts off full (combat movement uses no supplies).