Amount of water for starting boil.

Ok so the book I use as a guide for homebrewing calls for 4.5-5 gallons of water and bringing it to a boil then removing it from heat and adding LME/DME, hops, etc. But in other books they use half the amount of water and top off the carboy with cool water to reach the desired 5 gallon mark. Is there a difference in terms of quality with the beer in the end?

Yes, it's best to boil as much as your setup is capable of. Keep in mind you'll lose about a gallon during a good hour long boil.

You can make good beer by only boiling 2-3 gallons and then topping off to 5, but your hops utilization is better with a larger boil. With a partial boil you can also hold off on adding some of the extract until the last 10-15 minutes, this helps with the hops utilization and also helps keep your color lighter.