Short answer - Climb Speed is not the same thing as Climbing, and so Quick Climb and Boots of Speed don't really combine as you might expect for most of the game.
Without a Climb Speed, the actual distance you move when climbing depends on your Land Speed, as found in the Athletics skill. The majority of player characters will fall into the 20-35' speed category, and so will climb 5' or 10' normally (depending on degree of success). Note that this isn't actually a Climb Speed, so the Boots of Speed don't actually interact with it at all. It is simply the distance you climb if you don't have a Climb Speed. The Boots will increase your Land Speed, which might move you into another category of Climb movement.
Quick Climber doesn't actually alter your Climb Speed at all. Instead, it allows you to move an extra 5' or 10' (depending on degree of success) when Climbing, up to a maximum of your land Speed. So, given the expected 20-35 Land Speed, Quick Climber will increase your Climb move to 10' for success or 20' for Critical Success.
Boots of Speed add 5' to both Land and Climb Speed as an Item Bonus. So, if you have a Climb Speed (which is not the same as the movement you get from Climbing, even though this might seem counterintuitive), the Boots of Speed will increase that by 5', and Quick Climber will do nothing for you.
Normally, the only way the Boots of Speed and Quick Climb will interact is if you don't have a Climb Speed, and the Boots bonus to land speed move you into another category on the Climb Distance table (which could happen if you were an Elf with the Nimble Elf feat, for instance, or were a Monk with Incredible Movement).
This all changes a little if you have Legendary proficiency in Athletics. At this point, the Feat and Item will combine. Quick Climber will now give you a Climb Speed equal to your Land Speed. The Boots increase both Land and Climb speeds, so while your Land Speed will increase by 5', your Climb speed will effectively increase by 10', as far as I can see. This is not actually stacking two item bonuses to the same thing. Instead, it is applying different bonuses to different things, which then happen to combine. It would be similar to an effect that both increased your Constitution, AND gave you extra hit points at the same time.