One wants the $Q$-construction to have the property that $\pi_1(QP)=K_0(P)$.
To get this, one wants, for any covering space of $BQP$, that $K_0(P)$ acts naturally on the fiber over $0$.
To get this, one wants to associate to any monomorphism $i$ in $P$ a morphism $i_!$ in $QP$ and to any epimorphism $j$ in $P$ a morphism $j^!$ in $QP$ in a way that satisfies certain simple properties; the statement of the properties, and the proof that they suffice to get this result, is in Quillen's Algebraic K-Theory I (Theorem I).
Quillen's $Q$-construction is the Universal construction yielding such $i_!$ and $j^!$. (The proof is in QUillen's paper, immediately preceding Theorem 1.)
Therefore, there's a sense in which Quillen's $Q$-construction is the natural first guess for what should work. (Of course the "naturality" of this guess appears only in hindsight; a lot of other people failed to find this construction.)
PS. After you work through the constructions, you see that this is another way to see the same thing: For any object $A$ in your category $P$, you want to associate the $K_0$-class $[A]$ to some loop in $BQP$. The simplest thing to hope for is two canonically defined maps from $0$ to $A$ in $QP$, which together give you your loop. Quillen's construction provides those two maps (recognizing $0$ as a quotient of both $0\subset A$ and $A\subset A$) in the simplest possible way.