Hi, Well this is a tough one because the common name moon flower is often used in the Midwest to describe two different plants. The moonflower I know is a tropical vine, closely related to morning glory that develops dinner-plate sized white flowers that open at night. These are annuals and should be planted in the spring. The young plants have large leaves and look like a vine.
The other "moonflower" is actually called datura and is related to jimson weed. It has white, trumpet like blooms that are very fragrant. The seed pods are large and thorny. This plant is also an annual and should be planted in the spring. It, however, often re-sows itself if it's been planted in the garden the year before so some folks think it is a perennial--it's not. So, if you haven't had this plant before, you need to plant in the spring. Also, be aware it's very poisonous so don't let little kids mess with the seeds or seed pots. The leaves are sort of a dusty gray/green and the plant is very weedy looking (it is a weed in many locations)
I don't know which one you have, but both need to be planted in the spring.