As other have said: Quad Cannons are 'viable', just not 'optimal'.
Their main drawbacks are their abnormal firing cycles and [dmg] mod. They fire closer to non-heavy dual cannons, so they can trip up power flow and firing sequence. Powerful can be compensated for these days by just overcapping power and reducing weapon drain; firing sequence is purely a game/server issue - mashing spacebar to brute force it has its own drawbacks.
Their subjective pros, on the other hand, are the faster projectile speed and distinct sound/visual effects. The former means they're very slightly quicker at tearing apart things like fighters and seeking projectiles or just applying their actual damage to the target sooner(hit, miss, and how much damage is still applied the moment the weapon fires, just like every other weapon).
Personally speaking, I find they're fairly decent on battlecruisers when you want to add a little more forward bite to a ship - particularly beam array setups where you already have ample side-coverage and want just a touch more focused firepower without interrupting BFAW spam.
Agreed here. I have them on most of my Phaser builds as the only cannons (taking the place of Dual Beam Banks), and where seating allows I use Rapid Fire for them, which helps when I focus on the one ship.
"You don't want to patrol!? You don't want to escort!? You don't want to defend the Federation's Starbases!? Then why are you flying my Starships!? If you were a Klingon you'd be killed on the spot, but lucky for you.....you WERE in Starfleet. Let's see how New Zealand Penal Colony suits you." Adm A. Necheyev.
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Agreed here. I have them on most of my Phaser builds as the only cannons (taking the place of Dual Beam Banks), and where seating allows I use Rapid Fire for them, which helps when I focus on the one ship.