What plants will grow under a conifer hedge?
Lots of conifers create (as others have said) a very shaded canopy for whatever grows under them. And some make the soil more acid as time goes by. But some things naturally do grow under them. I suggest that you look for these things growing under similar species in the wild. Many mosses do grow under conifers, and do make a very nice carpet if you can get them to grow. Watering them generously, (and particularly during their establishment in a new site) will help them to begin feeling at home under your hedge, and will also cover any chemical abnormalities that the new site may have and which might cause the new moss to abstain from growing (or even establishing) at all. They often grow best in a thick carpet of their peers. Adding some humus underneath as a base layer might aid in establishing a strange moss. This might be worth a try – don’t just dismiss it as too unusual to try.
Nothing living, however, if you wish to look at a different side of gardening, why not start a fungus garden? Get a few logs of different trees, and put them under the hedge. Go to your local woods several times in the year to see what fruiting fungi there are, and collect a few mature specimens and put them amongst the logs. Water occasionally, but wait to see what might develop. Some of the fungi that grow on decomposing wood are really interesting and exciting – a conifer hedge is , admittedly dull but if you are stuck with one – make the best of it.