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 show is that the relative density of the set of primes represented by a positive binary quadratic form exists (I have no idea how hard this might be).

On the other hand, if you only want to know about a.p.'s of primes 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).

Edit: {The answer to your question, "...do we already know that a positive binary form represents arbitrarily long arithmetic progressions?" is yes. See the second paragraph below.}

If the relative density exists, so does the Dirichlet density and they are equal. The converse is not true in general. For primes in a given arithmetic progression, both densities exist. See Lang's Algebraic Number Theory, Ch. VIII.4 and XV. Given those facts, one approach to the problem would be trying to show that the relative density of the set of primes represented by a positive binary quadratic form actually exists (I have no idea how hard this might be).

On the other hand, if you only want to know about a.p.'s of primes 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).

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).

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 show is that the relative density of the set of primes represented by a positive binary quadratic form exists (I have no idea how hard this might be).

On the other hand, if you only want to know about a.p.'s of primes 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).