$T$ is not recursive.
Let $\Theta$ be the set of Gödel numbers of the theorems in $T$.
We have that $T$ is a consistent extension of $Q$.
By Set of Gödel Numbers of Arithmetic Theorems Not Definable in Arithmetic, $\Theta$ is not a definable set in $T$.
Suppose $T$ were recursive.
Then by Gödel Numbering is Recursive, we could recursively go from $\Theta$ to $T$.
Then using the recursivity of $T$, we could then recursively go back to $\Theta$.
Thus $\Theta$ would be recursive.
This would be a contradiction.