Is Cholesky the same as QR for this matrix? For a symmetric-positive-definite matrix $A=\begin{bmatrix}
a &  b\\
b  &  c\\
\end{bmatrix}$ with $a\geq c$ and eigenvalues $\lambda_1\geq \lambda_2 > 0$ can we say that Cholesky factorization with a lower triangular form is the same as QR factorization?
Actually I want to prove that $A_k$ in the following iteration converges to $diag(\lambda_1,\lambda2)$: for $k=1,2,...,$ $A_{k−1}=G_kG^{T}_k$; $A_k=G^{T}_k G_k$; end. I wanted to relate this to the power iteration if that's the way ...How to proceed?

No. The only orthogonal matrices that are at the same time triangular are diagonal matrices with $\pm 1$ on the diagonal.
What you can do is to compare the Cholesky decomposition or the more general LDLT decomposition with the LU decomposition. There the triangular matrices differ by a diagonal matrix.

Here you can directly calculate the QR decomposition, as $Q$ in the Givens rotation variant is
$$
Q=\frac1{\sqrt{a^2+b^2}}
\begin{bmatrix}
a &  -b\\
b  &  a\\
\end{bmatrix}
$$
while in the Householder reflection variant it would be
$$
Q=\frac1{\sqrt{a^2+b^2}}
\begin{bmatrix}
a &  b\\
b  &  -a\\
\end{bmatrix}
$$
