Convex Optimization with $ {L}_{1, 2} $ Regularization Term
The problem is given by:

$$\begin{equation}
\arg \min_{X} \frac{1}{2} \sum_{k} {\left\| {T}_{k} {X}_{:, k} - {Y}_{:, k} \right\|}_{2}^{2} + \lambda {\left\| G X \right\|}_{2, 1} \\ = \arg \min_{X} \frac{1}{2} \sum_{k} {\left\| {T}_{k} {X}_{:, k} - {Y}_{:, k} \right\|}_{2}^{2} + \lambda \sum_{l} {\left\| G {X}_{:, l} \right\|}_{2}
\end{equation}$$

In the above the MATLAB notation of : is used to select a column.
It is also assumed that the Mixed Norm $ {\left\| \cdot \right\|}_{2, 1} $ is operating on each column. In case working on each rows is needed one could easily transpose $ X $.

One could see above that the problem can be solved per column of $ X $ independently. Hence it can be solved in a separable manner column of $ X $ as a vector:

$$ {X}_{:, i} = \arg \min_{x} \frac{1}{2} {\left\| {T}_{i} x - {Y}_{:, i} \right\|}_{2}^{2} + \lambda {\left\| G x \right\|}_{2} $$

Now this is a simple problem which can be solved using Sub Gradient Descent and its accelerated variants.

The OP also mentions that $ {T}_{k} $ is the DFT matrix which is a Unitary Matrix namely it preserves the $ {L}_{2} $ norm. So the problem could be written:

$$\begin{aligned}
\arg \min_{x} \frac{1}{2} {\left\| T x - y \right\|}_{2}^{2} + \lambda {\left\| G x \right\|}_{2} &amp; = \arg \min_{x} \frac{1}{2} {\left\| {T}^{H} T x - {T}^{H} y \right\|}_{2}^{2} + \lambda {\left\| G x \right\|}_{2} \\ 
&amp; = \arg \min_{x} \frac{1}{2} {\left\| x - {T}^{H} y \right\|}_{2}^{2} + \lambda {\left\| G x \right\|}_{2}
\end{aligned}$$

In this form there is no need to calculate the DFT of $ x $ each iteration.

Some of the Math derivation is given in my answer at The Sub Gradient and the Prox Operator of the of $ {L}_{2, 1} $ Norm (Mixed Norm).