Intuitively, the correlation matrix is symmetric because every variable pair has to have the same relationship (correlation) whether their correlation is in the upper right or lower left triangle. It doesn’t make sense to say the correlation between variables $X_1$ and $X_2$ is $\rho$, but the correlation between $X_2$ and $X_1$ is $\rho’\neq \rho$ if calculating a Pearson correlation (so correlation is symmetric).
Mathematically, correlation between two variables, $X$ and $Y$, is commutative: $Corr(X,Y)=Corr(Y,X)$.
In OP’s case, the correlation between Q1 and Q2 is the same calculation and therefore the same result as the correlation between Q2 and Q1. Therefore the correlation matrix will be symmetric.
There are more mathematical reasons and proofs why a correlation matrix of real valued variables has to be symmetric and positive semi-definite, but I’ve excluded them from this answer.