OK, so I found it. It was actually a recipe on allrecipes.com that had a review that mentioned it. I will put a link to the recipe. Go down towards the bottom of the reviews and look for one by FAIRDOSE. She said red sugared beets. I hope that will help you.!
~Heidi
http://allrecipes.com/Recipe/Red-Velvet-Cake-IV/Detail.aspx
ps. this was not the recipe I used though. I do not know if it is good or not.
Years ago red velvet got its color from a chemical reaction between the cocoa powder and the acidity in the vinegar and/or buttermilk that was used in those recipes. This reaction is less obvious with dutch processed cocoa that is commonly used now (this reaction is also takes place in traditional devil's food cake recipes, which is how it got its name). When this reaction does take place, however, it's more of a deep reddish brown, and not the bold, vibrant hue that most red velvet cakes are today.
I found this on Grannygears.net I have never tried it as I LOVE my regular old Red Velvet cake and would never try this recipe because I detest beets! But it got pretty good reviews so give it a test shot! Let me know how it turns out!!!! Good luck!
Red Velvet Cake (Beet Cake)
Filed under: Healthy Recipes â Deanna @ 7:32 pm
Red Velvet Cake
2½ cup flour
1 teaspoon salt
2 teaspoon baking soda
1 cup sugar
½ cup unsweetened cocoa
1¼ cup melted butter
4 eggs, beaten
½ cup milk
2 teaspoon vanilla
1 cup shredded beets (cooked or raw)
mix dry ingredients; mix wet ingredients & add to dry ingredients; add beets & mix; pour into greased casserole dish; cook at 350º for 40 min or until fork comes out clean; frost with cream cheese frosting when cake is cool; refrigerates well.
The Martha Stewart recipe for Red Velvet cake has a few tablespoons of vinegar that you mix with baking soda right at the end -- then you add that to the batter, and mix for ten seconds.
I did this cake a while ago, and amazingly, it was a deep red hue (but not a "fake" red). I did not add the red food paste, either. It was a very moist, light recipe, and kept well for several days.
You could search her site for it (I don't think CC allows us to directly copy recipes from sites, otherwise I'd post it). 
Ladies,
Thank you so much for the valuable information regarding the beet juice. I will definately give this a go come Tuesday and I'll post the results on Wednesday. I will use the cream cheese icing I always use. I can hardly wait to give it a try. Thanks again.
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