  It is well-known that a celebrated J\"{o}rgens-Calabi-Pogorelov theorem for
Monge-Amp\`ere equations states that any classical (viscosity) convex solution
of $\det(D^2u)=1$ in $\mathbb{R}^n$ must be a quadratic polynomial. Therefore,
it is an interesting topic to study the existence and uniqueness theorem of
such fully nonlinear partial differential equations' Dirichlet problems on
exterior domains with suitable asymptotic conditions at infinity. As a
continuation of the works of Caffarelli-Li for Monge-Amp\`ere equation and of
Bao-Li-Li for $k$-Hessian equations, this paper is devoted to the solvability
of the exterior Dirichlet problem of Hessian quotient equations
$\sigma_k(\lambda(D^2u))/\sigma_l(\lambda(D^2u))=1$ for any $1\leq l<k\leq n$
in all dimensions $n\geq 2$. By introducing the concept of generalized
symmetric subsolutions and then using the Perron's method, we establish the
existence theorem for viscosity solutions, with prescribed asymptotic behavior
which is close to some quadratic polynomial at infinity.
