I've recently started reading the book "The Magicians" by Lev Grossman. There is considerable mention of a land known as Fillory, which seems to be related to Narnia, from C.S. Lewis's series.. Is there an intentional relationship, and if so, what is it?
Yes. From an interview with Grossman:
The funny thing is, Lewis was notoriously sloppy as a world-builder. He liked to drag in whatever was handy — nymphs, fauns, wizards, Father Christmas, whatever — without much regard to internal consistency. For Christ’s sake, Mrs. Beaver has a sewing machine! It drove Tolkien crazy. Part of the joke of The Magicians is that I’m taking a Narnia-style fantasy world and forcing it to behave consistently. It turns out that you have to bend it and distort it and break it to make it fit.
The Magicians essentially takes a re-look at two of fantasy's most popular tropes - the magic boarding school and the children off to fantasy land. The latter is exemplified by Narnia, and Grossman seems to have based Fillory off Narnia. So you have a bunch of children who go to Fillory, and fulfill the quests provided by the god-sheep.
However, as you read further you will find that though Fillory does resemble Narnia on the surface, it is given a much more adult treatment. People still die, and they seem to have more realistic personalities etc. The biggest difference is that unlike the books that follow the two tropes in a traditional manner, happiness is not guaranteed, and magic does not automatically make people happier or better.
Fillory plays the same role in the characters' childhoods and imaginations as Narnia does in our world, but it doesn't have a bunch of copyright lawyers potentially issuing cease-and-desist notices to authors who write about it.
I believe The Magicians borrows a lot of fantasy ideas from Narnia:
The magic wardrobe --> The magic clock.
The white witch, not native to Narnia but bent on controlling it --> The Beast/Mothra, not from Fillory but bent on controlling it.
The "land between lands" comprised of pools that act as portals to different worlds --> the fountains.
The concept of 2 kings and queens of Narnia, sons of Adam and Daughters of Eve (high king Peter, etc) --> 2 kings and queens of Fillory, both human also but without alluding to Christian themes as C.S. Lewis did.
The first to enter and explore Narnia is a young girl and boy, primarily the girl, Lucy, followed by her brother Edmund. --> the first in Fillory is likewise a girl and boy; Jane then Martin Chatwin. Also siblings, similar ages to Lucy and Edmund.
In both, time passes differently in the real world and in the Fantasy, and the similarities go on. Therefore, it's safe to say several parts of "fillory" were based on Narnia.