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.
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.
In my view, The Magicians essentially combines the ideas of Alice and Wonderland as well as Narnia and takes a Grimm Fairy Tales spin on it, at least in regard to Fillory plots. This is done by symbolism and similarity.
The first book of the Narnia series actually does start with a magician, two children, and a series of pools. Within the Narnia settings the pools are muddy and spread amongst a forest, but each pool led to another world... much like the pools leading to Fillory and other such places. The God Ember is a satyr much like Mr Tumnus in book 2. Later in The Magicians, the fairies appear and they are not unlike the white witch and followers within Narnia. The 4 rulers from Earth. The list goes on.
And like Alice in Wonderland, there are rabbits which serve as messengers between worlds. The air is opium, referring to the fact that Lewis Caroll was said to have written alive in wonderland based on drug induced hallucinations. They dig at Lewis Carroll's rumored paedophilia in regard to Alice by doing the bit with Martin Chatwin and the author of the Fillory books.
What's nice about this is that you can interpret what happens however you'd like and it would still be correct. You can take everything at face value, that this is about magicians where magic happens and there is a magical land with it's own lore and secrets, and it is enjoyable. Or you can look into it and interpret meaning from the objects and story lines, and it is easily just as enjoyable.