I realize this may be one of those questions where it's ill-posed and can't be answered without further constraints. But if not, what's the distribution like? In particular, is every integer equally often represented?
On the one hand, that appears likely based on the observation that regardless of what effect $\{a,b,x\}$ might have, $c$ can serve to exactly map one $\{a,b,x\}$ tuple to every integer. On the other hand, this feels suspiciously like one of those things that turns out to depend on how exactly you enumerate $a,b,c,x$ as you walk towards infinity.
I am also suspicious because of how I arrived at this question, via basic thoughts on the infinitude of quadratic primes. I reasoned as follows:
Suppose there are only finitely many quadratic primes, and let $f(x)=ax^2+bx+c$ be some function that returns primes for finitely many values of $x$. Let $m,n$ be the largest and smallest $x$ such that $f(x)$ is prime, and let $k$ be any integer outside of the interval $[n,m]$. Then $f'(x)=f(x+k)=ax^2+bx+c+ak^2+bk+2axk$ will be a quadratic function which is irreducible but always composite. Moreover, since there are infinitely many $k$ values yielding never-prime quadratics compared to finitely many $k$ that are sometimes prime, and the same would apply to any quadratic function used, it seems to follow that almost all quadratic functions must yield no primes.
And that brings us full circle, since if $ax^2+bx+c$ could be said to be "evenly distributed" in some meaningful way, and if my logic above holds, we'd have a contradiction proving infinitude of quadratic primes. Since we don't, quadratics must not be evenly distributed, and/or my logic above is flawed. If someone could point out where I've gone astray, I'll consider my question answered.
Edit:
Okay, let's do the naive thing and increase some $i\to\infty$ and consider $a,b,c,x$ in range $[-i,i]$.
While this is only a partial answer, I wrote a short program in Python to count how many times each integer is represented when $a,b,c,x \in \{-5,\dots,5\}$. (That's $11$ options for each value, meaning a total of $11^4$ combinations.) The following bar plot shows the occurrence of each value:
The most frequently obtained values were $1$ and $-1$ with $425$ occurrences. $0$ and wasn't far behind with $411$ occurrences. The lowest value reached was $-155$ with a two occurrences, and the highest value was $155$ with a two occurrences as well. (The plot is completely symmetrical.)
From this, my suspition is, even if we were to choose $a,b,c,x$ from a different interval, we would still find that some values are way more likely than others.
It may be possible to reverse-engineer what initial distribution of $a,b,c,x$ to use to get a near-uniform distribution, but that'd be way more difficult.
(EDIT: Corrected the plot and description. I accidentally used numbers form $-5$ to $4$, instead of $-5$ to $5$ previously.)
EDIT 2: I made a plot for the range $\{-10,\dots,10\}$ as well:
The distribution seems to tighten up around $0$ as we take larger and larger initial sets.