Now is a good time to take tip cuttings of this plant apply 0.8% IBA (rooting hormone) stick in well drained soil & mist. Bottom heat may help

'Plant Propagation' suggests a whip-and-tounge, chip-bud or T-bud graft onto rootstock. Buy some rootstock and give it a try, rootstock is only a few dollars each and if you do several you're bound to have one succeed.

Here is a link that might be useful: Fruit tree propagation

I am also trying to propagate a quince cutting outside. Should I bring it inside for the winter until it is established?

If you are taking cuttings and they are still inside I would recommend keeping them inside and then going normally from there in the spring. The root system will be to premature (not a healthy/large enough root system) and the the plant not hardy enough and will die if newly cultivated(which is why spring is the optimal time to do cutting as they have all summer to develop a root system).

They also need slow weather change outside to "harden off" and become slowly dormant for winter, and cant be simply grown in a warm climate and then dropped off outside in freezing temperatures and expected to survive (or at the very least have some extensive die back and damage). I had some Azalea cutting that took to long to cut and I myself have to keep them indoors for the first winter.

-Good Luck

pizza - I got the impression that the question was about flowering quince i.e. Chaenomeles, rather than true quince i.e. Cydonia, so I would go for cuttings as being much easier than grafting. For the latter you'd need a Chaenomeles rootstock and I'm not sure where you'd get one if you were not in the nursery trade or had time to grow your own from seed.

Flora_uk...Thanks for responding to this old post...actually, it is a true (fruiting) quince. As it turns out, we couldn't sell our house this year, so I have the opportunity to deal with the quince in the winter/spring...my time this summer and fall was taken up by propagating and then winterizing my in-ground fig.

Geoff