Mature seeds can germinate where they fall if they have proper water, soil, sun, etc. conditions. But the seeds on your grass may not be mature, so they won't be able to germinate. You could wait until the seeds are ready to fall off and then shake them into a bag and seed the areas you want, but it might be easier to buy a bag of quality grass seed. 

"No mow" isn't recommended for several reasons. First, it is really hard on your lawn to cut more than 1/3 of the height with each mowing. If your grass is tall enough to seed, it is likely too tall to mow in one pass, both for the health of your lawn and the ability of your mower.  

Second, the science behind "no mow" has been discredited. See https://www.ag.ndsu.edu/news/columns/dakota-gardener/dakota-gardener-the-messy-and-controversial-science-behind-no-mow-may for more info on "no mow". 

Thanks Terry for your response. That’s a bummer about no mow May as I enjoyed not mowing!