I have never done it,(using fleece as a backing) but I have sewed mittens out of fleece using a home machine, so I would assume you could quilt it quite handily!
Unlike Minkee, fleece isn't so stretchy.
when I made the mittens, I used a ball point needle and lengthened the stitches because of the thickness ... but, if you are free motion quilting, that probably wouldn't be an issue ... but the ball point needle in a denim (90/14 or 100/16) would be a good idea since I think fleece is knit, rather than woven.