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.
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. The spaces $W_\pm$ correspond to these finite dimensional vector spaces.