Yes you can. That's exactly what I am doing now. I have bitter melon seeds which are a year old. They didn't germinate after trials such as filing the hard shell of the seed, pinching the tip, soaking in luke warm water overnight... So brought a ripe bitter melon (the lady at the grocery store gave it away), cleaned the seeds under running water and am drying them as of today. Probably I might sow them tomorrow in a peat pellet. A few of them I wrapped them in a wet paper towel to sprout. I'll see if it is successful. It's the third day today.

Here is a link that might be useful: uj's vegetable garden

Yes u can..
here is what i did..
one of the melon became yellow so i did not want to throw the sseds..i have some from last year as well..
i wet them for 2-3 days before sowing..
The weather was in 80s for a couple of days last week..all the seeds are germinating now..
The weather should be really warm for them to germinate..