I have a Diophantine equation of the form: $$ax^2 + bx + c = y^2, \quad x, y \in \mathbb{Z^+}$$ Is it true that there will always be a linear recurrence formula that generates all the solutions for $x$, of the form: $$x_n = \alpha_1x_{n-1} + \alpha_2x_{n-2} + \alpha_3,$$ for some constants $\alpha_1, \alpha_2, \alpha_3 \in \mathbb{Z^+}$ (where plugging integers into $n$ generates the next $x$ that solves the Diophantine equation)?
If so, is it possible to prove this?
And if so, is it possible to show how these constants $\alpha_1, \alpha_2, \alpha_3$ are found?
I'm afraid this is overly optimistic. Multiplying through by $4a$ and completing the square, you go get a Pell type equation with some constant, and no more $b.$ The real problem is class number. The automorphism group of the Pell quadratic form is well behaved. However, there may be several seed values (pairs) rather than just two, and the rcurrence you describe takes you from one, well, fundamental region (cycle in Conway's topograph 'river') to the next. Read the first chapter in his Sensual Quadratic Form book and ask him about it.
I drew several topograph pictures for MSE answers, two are at Generate solutions of Quadratic Diophantine Equation . I drew on graph paper so the river is a perfectly straight line. That's life.