Many friends report finding seedlings around their Hydrangeas. I find some occasionally, but not directly underneath the shrubs. My plants are heavily mulched, which discourages seed germination. I have never attempted to propagate them from seed, because the progenies will usually be unlike the host cultivar.
I do root cuttings from some choice cultivars. Taking new wood cuttings in late summer has worked well for me. I use a 50/50 mixture of pine bark fines and sand or perlite, which allows sharp drainage of water to prevent rotting. Tenting of the pot with a plastic bag helps to prevent excessive drying. Use some type of support for the bag, to prevent it touching the cuttings. When kept moist, roots will usually appear within 2 to 3 months.
In my Zone, I can leave them outdoors to overwinter, but in colder areas, they should be protected from severe freezing temperatures.
Good luck with your seedlings!

thank you for your response georgiarose .... hmmmm, i am thinking that my little unexpected seedlings might be something i will regret giving *space* to in my garden ... when they don't come *true* does that probably mean they will be inferior? .... i think i will pull them out next spring ...

p.s. all the *cuttings* rotted, but i guess i *stuck* them at the wrong time of year? (spring)

thanks again, phyl

No, 'true' just means that they may not be completely identical to the parent. Flower size, flower color, plant development, winter endurance, height, growth habit, leaf form, etc, all those may differ slightly, a lot or none at all.