My experience is primarily with wild California hyacinths. Don't pick the seed heads when they are green. Wait til they get "papery" dry and somewhat gray or colorless. Then cut off the seed heads (unless you want the hyacinths to drop their seed "in place." Avoid waiting TOO long to remove the seed heads, because each tripartite seed chamber spontaneously opens when it is ready and drops its seeds, usually 3 per seed chamber, out.)

This next instruction is important: The seed heads must dry further, and be protected against moisture, which can encourage fungi, anaerobic bacteria, and spoiling. Best is to dry the seed heads in a paper bag in good ventilation, or spread out in a thin layer on a paper towel or plate on a sunny windowsill.

When Thoroughly dry, place the dry seed stems in a bag and shake the dookie out of them. Many seeds will come free of the stems and be easy to separate. You can pick out the remaining seeds if you wish, or simply toss the stems (with their remaining seeds) on the ground where you wouldn't mind having more hyacinths.

Allow the seeds to dry further before saving them however you usually do, ideally with a dessicant.

Even better, though, is to plant the seeds right away. They will sit tight in the soil (unless it is over-watered)til fall when slim green onion-like sprouts come up. My own summers are dry.

If they get water at the wrong time of year, they will sprout prematurely, but they tolerate that, they just don't bloom then. My sense is that they are only somewhat less "forgiving" than, hmmm, genus name escapes me, species=muscarini, "grape hyacinths."