It will only appear in the summer and you don’t need the seed to keep the grass growing, many varieties also spread sideways at ground level (the root bits grow knobbly lumps called called rhizomes).
So you can cut your grass and it will grow thicker if you cut it shorter more regularly (at least once a week and potentially twice a week if it’s growing fast) rather than letting it get long enough to set seed like yours as it encourages it so grow sideways if you keep cutting the tops off.
There are many different varieties of grass by the way, so the seed you buy is usually a mix depending on where you want your lawn to grow and what you want it to look like (shady or tough so kids can kick a foot vs bowling green)