Hello Marie
White clover, Trifolium repens, spreads by stolons and somewhat by seed, if flowers are pollinated by bees and the flowers are allowed to brown. Browning or ripening of seed takes 25 to 30 days.

In your lawn you will get more spread by stolons than from seed, since allowing the seedheads to ripen would also allow your grass to grow very tall, shading the clover and reducing the clover plants’ vigor.

Here are some facts about Trifolium species from a University of California article about growing the plant as a seed crop:

“Miller et al. (1951) recommended harvesting for seed when 90-95% of the visible heads are brown and the flower stems (seed stalks) have started to dry. In the Sacramento and San Joaquin Valleys, this would be in July and August. Withhold irrigation for 4 to 10 days prior to harvest. According to Duke (1981), seed should be harvested when most seed heads are light brown, about 25-30 days after peak blossoming. Mow and cure in swaths or in small windrows and retrieve using combines with pickup attachments. Each handling causes some shattering, so minimize the number of operation. Complete drying of seeds may require artificial means or spreading and turning the covered seed.

White clover revives quickly after mowing because stems are not cut. Four or five cuttings per year, at 35-40 day intervals, can be harvested in some areas (McLeod, 1982).

As related by Turkington (1989), Trifolium repens can reproduce either vegetatively or by seed?

In managing mixed stands of white clover and grasses, grass height must be controlled to allow white clover to do well (Carlson et al., 1985). Frequent mowing will encourage the clover (Miller, 1984c). Mixtures of grasses and white clover are difficult to maintain because grasses may shade the clover and because the clover requires high soil moisture levels. If soil nitrogen is high, grasses will tend to predominate, whereas the clover will dominate if nitrogen is low (Miller, 1984c)” 
The above is from—

https://sarep.ucdavis.edu/covercrop/whiteclover