Can a fat transfer be made to the throat region to obscure a prominent adam's apple?

Hello Happycamper94,
While your answer is yes, that would not be the best approach to obscure the thyroid cartilage (Adam's apple). 

Shaving the tracheal prominence surgically is a much more refined and predictable procedure than fat grafting, which may create irregularities and fullness in the neck.  With modern techniques, a tracheal shave is safe and the scar is easily camouflagable.  

Tracheal shave is the preferred method of augmenting a prominent Adams apple.  Fat grafting would not predictably correct a prominent laryngeal prominence and would likely create excess fullness in the mid neck. 

Fat transfer will likely just create a fat, unattractive neck with a strange bump in the middle.Adam's apple reduction takes 20 minutes and is remarkably safe when done by an experienced surgeon.

I would not recommend doing fat injection as opposed to a trach shave. Reducing the Adam's Apple is a safe procedure with limited downtime in the right hands. If you are wanting to avoid any risk to the vocal cords you should go to an experienced surgeon who looks at the vocal cords with a camera during the procedure. 
Fat transfer is a great procedure for the face, but it often takes multiple procedures to get the desired result. Injecting above the adam's apple enough to camouflage the bump will likely result in a full neck that blends into the chin which is not aesthetically ideal. 
Best,
Sarah Saxon, MD