Symmetric Cournot equilibrium: suffciency without second order conditon
How can it be a minimum or saddle since for every $q_2$ the profit function is strictly concave? If anything, the implicit non-negativity constraints might be binding, for example, if $q_2\geq 1/2$. 

When the objective function is strictly concave, the first order conditions are necessary and sufficient, and the maximum is unique. 

Not sure if I'm answering your question, but hopefully it helps.