What fruit does Vermont produce?
Apple is the official state fruit for Vermont.
No, a lilac bush does not produce berries. Specifically, the lilac bush (Syringa vulgaris) does produce a fruit. But its fruit does not fit in with the definition of a berry as the fleshy, simple fruit which is produced by one ovary. Instead, its fruit meets the definition of a capsule as a dry, simple fruit which is made up of at least two carpels (ovules) within which are the plant's seeds.
The tree that you have in the beginning of the story does produce fruit. If it produces fruit in Summer, then it will produce flowers in Spring. Then the other way around would be if Spring produces fruit, then Summer will produce flowers. Anyway, if you don't want to wait that long, then go buy tree seeds from Vesta's shop. Then just plant them, and you don't need to water them, just wait.