I think you can, as they sometimes seed themselves, when you find a seedling growing. One year I had a really good plant from a chance seedling. Like a lot of plants, if you try to grow them, they damp off on you or fail to germinate.

I think surfinias are F1 hybrids, which means they produce no seed, or sterile seed, or seed which gives other results, in other words, any germination from flowered plants you've bought won't be actual surfinias. They sometimes survive the winter outside in a very sheltered spot though.

F1 Hybrids often produce seed, but the seedlings reshuffle the genes of their parents in such a way that only a minority of the seedlings are very much like their parents. As Bamboo says, what you get is unlikely to be an actual 'Surfinia'. With these, and 'Wave' Petunias, and 'Supertunias', commercial growers often keep plants growing in greenhouses over the winter, and start cuttings early the next spring, to preserve the best strains.

'Surfinias are produced from cuttings under copyright, and seeds can't be purchased commercially,however by saving seed after flowering, one can sow these in Feb/March.' Found this info on 'over the garden gate' website.