In General linear group $GL(n,q)$, every matrix is conjugate with its rational canonical form. Using this fact we have the conjugacy classes of cyclic groups. My question is about the groups which are generated by two elements. I tried to find some results about conjugacy classes but it seems to be hard, any comment is so appreciated.
You would have an answer to your question if you could classify pairs of elements in $GL_n(\mathbf F_q)$ up to simultaneous conjugacy. This is the notorious matrix pair problem, which is the quintessential wild problem in representation theory (see When is a classification problem "wild"? , Are wild problems related to undecidable ones? and How can classifying irreducible representations be a "wild" problem?). So you are unlikely to get a nice answer for general $n$.