A partial answer only. Dealing with the case of an odd $b$.
Assume first that $a$ is even.
Let us consider the polynomial function $f(x)=ax^2+bx$ from the residue class ring $R=\mathbf{Z}_{2^m}$ to itself. I claim that this function is bijective. As the ring $R$ is finite, it suffices to show that $f$ is injective. To see this consider the possibility that
$$
f(x)\equiv f(y)\pmod{2^m}.
$$
This means that
$$2^m\mid f(x)-f(y)=a(x^2-y^2)+b(x-y)=(x-y)(b+a(x+y)).$$
Here the second factor $b+a(x+y)$ is always odd, as $a(x+y)$ is always even, and $b$ was assumed to be odd. So for $2^m$ to divide $f(x)-f(y)$ it is necessary (and obviously also sufficient) that $2^m\mid(x-y)$. But this proves our claim.
From the bijectivity of $f$ it follows that the solutions of the congruence
$$
f(x)\equiv -c \pmod{2^m}
$$
form a single residue class modulo $2^m$ irrespective of the value of $c$.
Let us then assume that $a$ and $b$ are both odd.
Clearly $ax^2+bx$ is then always even, so for this equation to be solvable we must have $2\mid c$. I claim that a solution $x$ (in fact two distinct solutions) always exist for any even $c$.
To see this let us study the function $p(x)=ax^2+bx$. Consider the possibility that $p(x)=p(y)$ for some elements $x,y\in R$. This is equivalent to
$$
2^m\mid p(x)-p(y)=(x-y)(a(x+y)+b).
$$
Here always $x-y\equiv x+y\pmod2$. As $a$ and $b$ are both odd, this implies
that the two factors, $x-y$ and $a(x+y)+b$, have opposite parities. Therefore it follows that $2^m$ must divide one of them, and we can conclude that either $y\equiv y_1\equiv x\pmod{2^m}$ or $ay\equiv-b-ax\pmod{2^m}$. As $\gcd(a,2^m)=1$, the latter congruence has a unique $y_2\in R$ as a solution. Furthermore $y_2\not\equiv x\pmod{2}$, so $y_1\not\equiv y_2\pmod{2^m}.$
So we see that as a function from $R$ to itself, the polynomial $p$ attains all the values in its range exactly twice (two-to-one). Therefore $|p(R)|=|R|/2$. As the range is a subset of even residue classes, we can conclude that $p(R)=2R$.
As a conclusion we can say that when both $a$ and $b$ are odd, the original congruence has two non-congruent solutions, if $c$ is even, and none if $c$ is odd.
