It's normal, and you're right to mow your lawn normally. But don't expect the seed to germinate. Most of the grass cultivars we use today are sterile, so they don't breed from that. Homeowners can rest assured that the grass to be planted is perfectly healthy and is the natural process for the lawn to reproduce itself.
As unsightly as it sounds, there is no real way to prevent grass from becoming seed during this time. The grass you are going to sow is a good sign that the plant is really healthy and grows well. If it's really worrying, the only way to minimize the appearance of grass seed is to cut it continuously to remove seed heads. The cold-season lawn goes through a mid-spring process in which it blooms and begins the process of sowing seeds.
These immature seed heads make the lawn look “feathery” and can detract from the overall dark green color of your lawn. The flower is born on a stiff stem that rises above the leaves and is usually present for 2-3 weeks depending on the type of grass present. This stem is hard and difficult to cut cleanly and can leave an uneven end even when the mower blades are very sharp. It is a natural function of the grass plant and is its way of trying to keep surviving.
Every year, usually from May to mid-June, many lawns begin to show signs that grass is going to be sown. When these grass seed heads start to appear on the lawn, it is understandable to think that you may have weeds. Even if the seed was allowed to reach maturity, which would take about four months, allowed to dry and then harvested, you would still have to make sure that the seed would find a home in the ground to be able to germinate. The most common types of grass in Australia produce a sterile seed head, which means that they cannot be spread by seed in other areas of the garden and grown from the seeds, only through twigs or vegetative stolons.
It is very easy to confuse Couch seed heads with weeds, since their color varies from green to purple and they grow above the level of the grass.