There are several recipes with the beet juice under titles about pickling quail eggs. I finally did it and pickled some this morning. I used 1/2 apple cider vinegar and 1/2 water. I put salt to taste in each jar instead of in the vinegar solution, used a little more than the 1/2 tsp per pint that you can veggies with. I put 1 clove of garlic, sliced jalapeno peppers, slivers of red pepper for color, whole peppercorns, dill weed and dill seed, celery seed and mustard seed in each jar. I did some of that as much for color and texture as much as taste. They look good but it's too soon to try them yet.
The only thing I would do different since they were quail eggs is do it on a Sat. and invite select grandchildren to help with the egg peeling part. I'd choose the boy with proven talent in this field and the girl who is a primary consumer of eggs..(or maybe not her, she'd have a hard time leaving enough to pickle).
Good luck on you search, hope we both come up with something yummy.