As Amzing says, any machine can be used for quilting, though obviously some are better than others. You can get a 1/4" foot to get accurate seam allowances but you can also set that up with your normal foot if you can adjust the width on the straight stitch (this has the effect of moving the needle left & right). A walking foot is reccommended for when you 'quilt' the layers together as it moves the top layers as well as the bottom and helps stop it puckering. If you want to do free motion quilting then you can get a foot for that and you need to be able to drop the feed dogs.
There are machines with a lot of space between the needle and the side of the machine, which makes things a lot easier BUT they tend to be siginificantly more expensive.
yes, you can quilt a king size quilt on the average sewing machine, need to roll up tight the bits under the arm. best if you look in a shop to see what you like the look of and what you can afford.
if you can find a place where sewing machines get fixed they can give you brilliant advice and you can buy used but re-serviced machines for much less than brand you (half or quarter of original price!)
there are quilt/fabric shops everywhere who also sell machines. and there are quilt shows where you can also look and try out and get expert advice.
I actually prefer doing the patchwork part of quilts, the quilting itself is more a practical thing for me, so I keep it simple. I sometimes plan to try some elaborate feather motif or vermicelli or hearts but end up reverting back to straight lines so I can finish the quilt and use it!
can you borrow a sewing machine to do a small project to start with?
WRT your question about whether or not a quilt would fit into a sewing machine, as others have said, a tightly rolled up quilt should fit into most standard machines but you can, if you pay a bit more get machines with a longer arm with are designed specifically with quilitng in mind.
On the other hand, if you want to quilt a whole quilt you are best off avoiding the budget or very lightweight, cheap machines because they may have a shorter than average arm.
You can get some very complicated very advanced machines but I would give those a miss personally. They are impossible to cart around so if you join a group or go to a class you have no hope of moving them. Plus there is more to go wrong. I also find that I have a fairly advanced machine with a lot of embroidery stitches for example, but I rarely, if ever use them. So long as you can vary the length of stitch and do a zigzag and maybe a button hole you are sorted.
1/4" feet seem to be a mixed blessing. Some love them, some don't but I think it partly depends on how accurate your machine is with them fitted. Mine seems to be spot on (Janome machine) but I know others aren't. I suspect it varies from machine to machine rather than by make. You can also get machines where you can fit the 1/4" as a guide on the foot plate which is helpful too.
Personally I think you need to try out a few machines but don't be distracted by the whistles and bells, just see if you like the way it is set up.
I have a Janome Quilters Companion machine that my DH bought me for Christmas approx 6 years ago. It has many of the features bran has mentioned, has a quarter inch foot which I love and a walking foot which is brilliant for machine quilting. I also regularly use the zipper foot, and occasionally some of the decorative stitches.
I do an assortment of quilting, patchwork and hobby sewing.
When we bought it, I took a lot of advice from my patchwork tutor, and went to a specialist retailer who were brilliant. They run regular classes on how to get the best out of your machine, etc. and were very helpful when I thought I had a problem ( I didn't - it was me being clueless ).