If I have understood your notion of "binary language" (which is unfortunately not defined in the question), it appears that your question (in the special case $m=3,n=7$) is isomorphic to the following:
Is the language $K^\$ = \{b^{3k} c^{4k} : k \in \mathbb{N}\}$ regular over the alphabet $\{b,c\}$?
The answer to that question is "No". To see why, see the relevant reference questions here: Reference answers to frequently asked questions and How to prove that a language is not regular?.
Explanation:
The isomorphism is as follows: $b=(a,a)$ and $c=(\$,a)$. In other words, when you have the pair $(a,a)$ in the binary language, we can map this to a new symbol $b$ (in an ordinary, "unitary" language); when you have $(\$,a)$ in the binary language, we can map it to a new symbol $c$; and when we have $(a,\$)$, we can map it to a new symbol $d$. Now all words in the binary language $L^\$$ are a concatenation of some sequence of $(a,a)$, $(\$,a)$, and $(a,\$)$, so we can map them to a word over the alphabet $\{b,c,d\}$.
Or, to put it another way: if I've understood the definition of a binary language correctly (which I might not have), you can think of the language $L^\$$ as a binary language, or you can alternatively think of it as an ordinary, "unitary" language over the alphabet $\{(a,a), (\$,a), (a,\$)\}$.
When $m=3,n=7$, applying the above isomorphism to $L^\$$ yields the language $K^\$$ I defined above. Now standard methods suffice to determine whether $K^\$$ is regular; indeed, for $m=3,n=7$, it is not regular.
The general question:
You should be able to use similar techniques to answer your question for all $m,n$, on your own.
Background on the site:
To give you a little bit of further explanation about this site: The expectation on this site is that you will work out the details of those on yourself. If your question can be answered using the techniques described there, then it is most likely not a good fit for this site. This site's primary purpose is to create an archive of content that will be useful for others. Also, we expect people to do some research/work on their own first before asking here. If the question can be answered using standard techniques (for proving that a language is not regular), then the question is not likely to be a good fit for this site, because it is likely too localized to be of use to others and because it does not meet the requirement for self-effort.
If this is not what you were asking, you'll need to edit the question to be more clear about what you are asking and to define all terms. You should also edit the question to show what you have tried so far.
