(A quick search shows lots of threads involving Quad Laser Cannons, but no topics that suggest an obvious answer contained within. So apologies if this has come up before...)

I have 2 energy on the Quad Laser hard point card.

My first attack hits, thus I cannot spend the second energy from that weapon on the target.

But can I elect to use the same weapon to attack a second target with the remaining second energy token?

NO

NO

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I never have, and something tells me that an individual weapon card can only be used once per round. But is there an explicit rule stating that? Quads are the only currently available hard point where it's energy capacity even makes two attacks a theoretical possibility.

During the Combat phase, each huge ship may perform one attack with its

primary weapon and may perform one attack with each of its secondary
weapons. Each attack must be fully resolved before it begins another.

Quads are a slight exception, but their wording is similar to cards like cluster missiles and not gunner. Cluster missiles says 'this' while gunner says 'another'

Huge Ship Rules!