I have the same initials, and I agree with TootsNYC - I would find it odd to receive a thank you note with Esq attached.
(You are talking about Esq, right? Not J.D.? Because J.D. would be inappropriate in any case.)
Generally, as far as I'm aware, Esq. is only used out of respect when corresponding with someone IN THEIR CAPACITY as lawyer - corresponding on the litigation or deal going on...