If the relative density exists, so does the Dirichlet density and they are equal. For primes in a given arithmetic progression, both densities exist. See Lang's Algebraic Number Theory, Ch. VIII.4 and XV. Given this, all you have to do is show that the relative density of the set of primes represented by a positive binary quadratic form exists. 

However, if you're only interested in a.p.'s represented by a positive quadratic form, a better approach might be answering the question, "Does Green-Tao still hold for sets of primes with positive Dirichlet density?" The answer is yes since G-T only requires that the limsup of the relative density be positive, and positive Dirichlet density implies positive limsup (if the limsup were 0, the lim would be 0).