Hi CMK,
The seeds will not come true. Any shrub that is not a straight species will have variations in the offspring even if it was pollinated by itself. People might be interested in growing your seeds though. Moned is a red cultivar and red is common with quinces so the offspring will most likely be red. It seems Moned is marketed for its big flowers so seedlings might have smaller flowers.
Quinces are in the rose family and I would treat the seeds the same for highest germination which is to keep moist in the fridge. I personally would just keep the whole fruits in fridge and then open them as you need to send out to a trader. I've done this before for Key Limes. But now that I think about it, you do need to trade these out in the next couple months. They need to be stratified so winter sowing is best. Oh and make sure your fruit are really ripe for mature seed.
Hope this helps,
Remy

You are wonderful Remy! Thanks for the info.

Should the fruit be very soft for the seed to be mature? Mine were hard. They also fell off of the plant a week or so ago, while some stayed on. What does perfectly mature seed look like? Right now, they are medium-dark brown, just like an apple seed. If you think these are not mature I won't trade them.

As for storing moist/cold...is storing the seeds in a baggie between damp paper towels a good method?
CMK

Hi again CMK,
Thanks! That the nicest thing I've read all day.
No the fruit should be hard. Quince fruit stays hard forever it seems. So people can inadvertently pick under ripe fruit because of that. If they are falling off and your seeds look like ripe apple seeds, they are good.
Storing between paper towels can work at least short term. I've mailed seeds that way with instructions to plant immediately and have had no problems. The paper is not sterile which could be a problem for longer storage since mold, etc. can grow. For longer storage putting seeds in with damp sterile potting medium or just plain vermiculite if you have some in little zip locks would be better. This is the way some seed places do it and the seeds from the NARGS exchange come this way.
Remy

You are a wealth of information!

Great! I will be sure to trade these right away, and tell everyone to sow ASAP. And lucky for me I had the foresight to buy extra potting soil last weekend!

Hope you have a wonderful day Remy.
CMK

As far as I know, there MUST be two types to set seed. Is it possible a neighbor has one or that you have two very similar types? These are so easy from cuttings that I get great results from just sticking them in the ground and watering them.