Quinn is a very good ADC, if played right she can own the game and lead her team to victory.
In my opinion, she is one of the best.

To start off you want to deal a lot of damage to survive longer because of the amount of health you have.
The Long Sword is the best for that, and you can buy 3 Health Potions so you can heal up when needed.
The Trinket is your choice but i prefer Warding Totem because you can see is anyone is hiding from you early game.

These items are to help reach your main two items earlier.

The Boots and the Dagger help out when getting the Greaves and the most important item, the BOTRK because mid to late game it deals so much damage, you start yelling "THE DAMAGE THOUGH!!"

Now as we mentioned before, the first two items are,

Berserkers Greaves
BOTRK

Now as we progress, we will have had enough kills or farm, or both, to buy, Youmuus Ghostblade, this item will provide that little extra help when up against that support like Leona or Taric who have armor and health.

Then we get to the big damage items.
The Bloodthirster is the item of legends, it is the queen to the BOTRK, why? Because of the damage and the life steal, which comes in handy against those tanks :)
And the after we have 10 or so kills we can get the Last Whisper which will deal damage and give the opponents something to worry about, if they aren't already running that is.

Now we have 4 choices here, and here is why,

The Guardian Angel is the most recommended item to use with Quinn as it provides some armor so you can take more hits and magic resist so you can deal damage to those mages with out receiving much in return.

The Frozen Heart is when ya up against the real tanks like Leona or Volibear and need to use more abilities quicker and more frequently.

The Warmog
s Armor is just in case you wanna go and piss people off, with the health and damage, i'm sure that's easy :)

And lastly the Spirit Visage is for when you need that extra life steal, cooldown, health and magic resist but can't buy two items. This is a really good one because it's passive, increases your life steal by 20%, which can come in handy if all a needed was that extra bit of health left.

Now this item is only allowed to be bought when you have finished ya build but aren't that close to winning, and here is why.

The Zephyr has the extra attack damage but still has the movement speed and double the attack speed, with some more cooldown, which could be the game changer, using your Q every 2 seconds or less!.

So with this item at very last, which you should only buy once you have all the money too, you could lead your team to victory! :)