I have no experience of doing that. But I do have a lot of fungi (mushrooms) in my garden.
I know that the mushrooms that we see above ground are the fruits of organisms which grow almost entirely beneath the ground, as a fine web.
The web gets the foods to nourish itself from other host plants. In most cases the fine web becomes part of the root system of the host plants. The fungus cannot exist without the host plants that it is feeding on. In the case of grassland, the mushroom may be feeding on the roots of the grasses. The fungus helps the host plant in return, by increasing the area of the roots of the host plant.
If you were to transplant a square of turf, which incorporated the fibrous web, then it seems to me reasonable that that could produce the fruiting bodies next season.
There are techniques for trying to grow cultivated mushrooms, in the dark on compost, or by injecting spores into dead wood. But growing mushrooms in grass seems far more natural, if it can be done.