Using Perov's fixed point theorem, the authors establish the existence and uniqueness of solutions to the coupled system of implicit fractional differential equations \[ \begin{cases} {}^{c}\!D^{lpha}x(t) = f_{1}(t,x(t),y(t), {}^{c}\!D^{lpha}x(t)), & tı J, {}^{c}\!D^{\beta}y(t) = f_{2}(t,x(t),y(t), {}^{c}D\!^{\beta}y(t)), & tı J, x(0) = L_1[x], \quad x'(0) = L_2[x], y(0) = L_3[y], \quad y'(0) = L_4[y], \end{cases} \] where $\alpha,\beta \in [1,2)$, $J=[0,1]$, ${}^{c}\!D^{\alpha}$ and ${}^{c}\!D^{\beta}$ are Caputo fractional derivatives, $f_{i}: [0,1]\times\mathbb{R}^3 \to \mathbb{R}$ are continuous functions for $i=1, 2$, and the functionals $L_{j}$, $ j=1,2,3,4$, are Stieltjes integrals. A second existence result is obtained by using a vector version of a fixed point theorem for a sum of two operators due to Krasnosel'skii. There is also a study of the structure of the set of solutions to the problem. Examples illustrate the results.