Bonsai trees are the same as trees in the ground except we keep them in pots and trim regularly.
Any plant needs to be physically mature to produce flowers. Some species can flower and make seeds at just a year old while some take many years to mature. Not sure how old Chinese elms need to be to produce seeds. I know the elms in my garden have seeds after around 10 years but could have been mature a few years before I noticed.
Being in pots won't stop them flowering and producing seed but the regular trimming often removes the flower buds before they are visible so a lot of our bonsai do not produce seed.
As
@erich_raudebaugh has just posted many growers trim off flowers or developing seeds thinking they look bad or will take vigour from the tree.
Some trees need pollinators to help spread pollen from one flower to another so they may or may not produce viable seed depending whether there are the correct insects or winds to help. A few species are dioeceous meaning some trees are males and some are females. Obviously the males can't produce seeds and the females need a male tree somewhere nearby to produce viable seed. Pretty sure that elms are self fertile so each tree should be able to make seed, even without another partner nearby. They are insect pollinated - the bees flock to ours when they are flowering - so the tree will need to be outdoors while flowering and have insects interested in the flowers to move the pollen from one flower to another.
Another hiccup can be tree health. To produce viable seed the tree needs to be healthy. Some unhealthy trees can produce seed but it is empty with no embryo it cannot germinate.
After all that it is possible for your Chinese elm to produce viable seed but in all the years I have been growing Chinese elms as bonsai I have never seen seed on any of my bonsai specimens so I would not pin all my hopes on that source. Look around your area for mature Chinese elm trees in parks, gardens or street plantings. They flower in Autumn and the seed matures in just a few weeks and falls before winter sets in.