How to riffle matrices

I would use Flatten:

m = {{M11, M12, M13}, {M21, M22, M23}, {M31, M32, M33}};
n = {{N11, N12, N13}, {N21, N22, N23}, {N31, N32, N33}};
p = {{P11, P12, P13}, {P21, P22, P23}, {P31, P32, P33}};
q = {{Q11, Q12, Q13}, {Q21, Q22, Q23}, {Q31, Q32, Q33}};

Flatten[{{m, n}, {p, q}}, {{3, 1}, {4, 2}}] //TeXForm

$\begin{pmatrix} \text{M11} & \text{N11} & \text{M12} & \text{N12} & \text{M13} & \text{N13} \\ \text{P11} & \text{Q11} & \text{P12} & \text{Q12} & \text{P13} & \text{Q13} \\ \text{M21} & \text{N21} & \text{M22} & \text{N22} & \text{M23} & \text{N23} \\ \text{P21} & \text{Q21} & \text{P22} & \text{Q22} & \text{P23} & \text{Q23} \\ \text{M31} & \text{N31} & \text{M32} & \text{N32} & \text{M33} & \text{N33} \\ \text{P31} & \text{Q31} & \text{P32} & \text{Q32} & \text{P33} & \text{Q33} \\ \end{pmatrix}$

♮ = ## & @@@ (#) &@(## & @@@ (#) & /@ (#) & /@ {##}) &;

♮[{m, n}, {p, q}] // MatrixForm