I think the missing link for you is not realizing that $Q$ is actually equal to $q_1 + q_2$: in a duopoly, market quantity demanded can only be derived from the two firms in question. With that in mind, we can rewrite the profit functions for both firms in terms of $q_1$ and $q_2$ and then optimize with respect to the $q$'s accordingly (since Cournot competition is competition over quantities). We are then going to be left with two best response functions, one that tells you the optimal level of $q_1$ as a function of $q_2$ and another that tells you the optimal level of $q_2$ as a function of $q_1$. Then it just becomes an algebra problem to find the Nash equilibrium and the relationship of quantities and profits between the firms will become clear.
Firm 1 maximizes profits
\begin{align*}
\Pi_1(q_1) = (\alpha - (q_1 + q_2))q_1 - c_1 q_1.
\end{align*}
The first-order condition w.r.t. $q_1$ is
\begin{gather*}
\frac{d \Pi_1(q_1)}{dq_1} = 0 \\
\implies \alpha - 2q_1 + q_2 = c_1 \\
\implies q_1(q_2) = \frac{\alpha - c_1}{2} + \frac{q_2}{2}.
\end{gather*}
Similarly, Firm 2 maximizes profits
\begin{align*}
\Pi_2(q_2) = (\alpha - (q_1 + q_2))q_2 - c_2 q_2.
\end{align*}
The first-order condition w.r.t. $q_2$ is
\begin{gather*}
\frac{d \Pi_2(q_2)}{dq_2} = 0 \\
\implies \alpha - 2q_2 + q_1 = c_2 \\
\implies q_2(q_1) = \frac{\alpha - c_2}{2} + \frac{q_1}{2}.
\end{gather*}
Solving the system of equations for $q_1$ and $q_2$ yields
\begin{align*}
q_1^* &= \frac{\alpha - 2c_1 + c_2}{3} \\
q_2^* &= \frac{\alpha - 2c_2 + c_1}{3},
\end{align*}
which implies that $\mathbf{q_1^* > q_2^*}$ since $c_1 < c_2$.
Note that both firms face the same price $P^* = \alpha - (q^*_1 + q^*_2)$. Therefore, we have that
\begin{align*}
(P^* - c_1)q_1^* &> (P^* - c_2)q_2^* \\
\implies \mathbf{\Pi^*_1} &> \mathbf{\Pi^*_2}.
\end{align*}
