I have non-square real matrix $X$ whose size is e.g., $2$ x $6$, I need to find the square real matrix $Y$ different than $0$ with size $6$ x $6$ such that $XY = 0$.
I have tried this using the Null space, which can be found in matlab as follows: Y = null(X), but that matrix $Y$ in this case is not square of size $6$ x $6$. Is there a way to get that square matrix $Y$ such that $XY = 0$ ?
If you've already found the null space, the next step is build a $6 \times 6$ matrix where every column is chosen from that space:
$$Y = \begin{bmatrix} {\bf y}_1&{\bf y}_2&\cdots&{\bf y}_6 \end{bmatrix} $$
where ${\bf y}_i \in \operatorname{Null}X$. Then, where $X = \begin{bmatrix}{\bf x}_1\\{\bf x}_2\end{bmatrix}$, we have $(XY)_{ij} = {\bf x}_i\cdot{\bf y}_j = 0$, as required.
If you don't care about finding all such matrices and only need to find one, then the simplest option is to make all the columns the same.
You didn't provide a specific example, so let me make one up. Suppose we have
$$X = \begin{bmatrix}1&1&1&1&1&1\\2&0&0&0&0&0\end{bmatrix}$$ Then the null space is $$\operatorname{Null}{X} = \{(0,-a-b-c-d,a,b,c,d): a,b,c,d \in \Bbb R\}$$ To pick an element from this space, choose any $a,b,c,d$. Let's keep it interesting by not choosing all zero; say $a=1, b=1, c=0, d=0$. This gives the vector $(0,-2,1,1,0,0)$. We'll use this as our first column. Let's take $a=b=c=0$, $d=-5$ for the next column. So far we have:
$$Y = \begin{bmatrix}0&0\\-2&5\\1&0&\cdots&\\1&0\\0&0\\0&-5\end{bmatrix}$$
and the remaining columns can be chosen similarly by choosing arbitrary values for $a,b,c,d$. Once again, you could elect to make every column identical. Try calculating $XY$ to see what happens.