You don't have to transplant, just thin them to 6" (13 cm.) spacing.
But if you need to transplant them to a better spot, the best time is right when they start growing in the late winter, early spring.
Here, we either direct seed in the place they will grow all season or start in a nursery bed (or flats) and transplant before the cold/wet sets in (late autumn).

Here in central California we grow our garden onions mostly from transplants. Your climate sounds similar to ours (light frosts). I just sowed about a 1000 seed of onions this morning (it is mid summer here). I allow the onion seedlings a couple of months to grow and transplant them about an inch apart and let them grow until I want to transplant them the final time in (about 6 inches apart) the garden in January/February (winter here). I double transplant to save room as my garden still has the summer garden growing in October. If I had the room I would single transplant in October.