In the special case $G= \mathrm{GL}_2(k)$, a more direct approach than those indicated in the answers and comments is certainly possible even though it doesn't do much to illuminate the general case.
To simplify a bit, note that the center (consisting of nonzero scalar matrices) is itself a 1-dimensional torus not conjugate to others; so it's safe to consider just the derived group $\mathrm{SL}_2(k)$. Here the 1-dimensional tori are just the maximal tori in the algebraic group setting, usually called Cartan subgroups in the more specialized Lie group version when $k= \mathbb{C}$ or $\mathbb{R}$.
In the Lie group case, you are asking for all conjugacy classes of Cartan subgroups in the real group. This is an old problem, solved in general by Kostant for semisimple Lie groups (and independently by Sugiura, available online here). It's equivalent to finding the conjugacy classes of Cartan subalgebras. In your special case there are two classes, represented by the diagonal torus and a compact version.
In the algebraic group case, you are similarly asking for all conjugacy classes of maximal $k$-tori in the case when $k$ fails to be algebraically closed. Here again there is a lot of general theory, organized by Borel-Tits, with the machinery of Galois cohomology then being invoked to discuss $k$-forms. But some short-cuts are likely in your rank 1 situation. What approach you take depends a lot on what your ultimate interest is and which fields are most important.
For higher rank groups, the whole problem has a different flavor. In the algebraic group setting, you'd be looking at the dual of the character group of a maximal torus consisting on co-characters (or 1-parameter subgroups) relative to a splitting field and its subfield.
