There is an altogether different motivation different from the ones discussed above that appears in a paper by Graeme Segal ("K-homology and algebraic K-theory," LNM 575 K-theory and Operator Algebras, Athens Georgia 1975, pp. 113–127).
The $Q$-construction there is motivated by considering self-adjoint Fredholm operators on Hilbert space.
More, precisely Segal shows that the homotopy type of the classifying space of Q-construction of the category of finite dimensional vector spaces over the reals or complex numbersis the same as that of the space $Saf(H)$ consisting of self-adjoint operators on infinite dimensional Hilbert space $H$: $$ BQC \simeq Saf(H) . $$ A map $V\to V'$ in the $Q$-construction on the category $C =$ Vect of finite dimensional vector spaces is represented as a triple $(W_+,W_-;\alpha)$ in which $\alpha: W_+\oplus V\oplus W_- \to V'$ is an isomorphism of vector spaces.
According to Segal, the idea is supposed to be that a Fredholm operator is determined up to contractible choice by its kernel and cokernel, which are a pair of finite dimensional vector spaces. When a Fredholm operator is deformed continously, its kernel and cokernel can jump but only by adding isomorphic pieces to each.
In the self adjoint case, the operator is determined by its kernel up to contractible choice. The kernel then corresponds to an object of the $Q$-construction. When the operator is deformed, the kernel jumps in such a way that the part added to it is a direct sum of on the part on which the operator was positive and a part on which it was negative. These correspond to the terms $W_\pm$ appearing above.
So the motivation in a nutshell is this: the objects of the $Q$-construction correspond to self-adjoint Fredholm operator kernels and the Morphisms correspond to deformations of such operators.