[Community Question] Linear-algebra: Calculating only the needed part of Q of thin QR decomposition

One of our user asked:

A rectangular, $A \in \mathbb{R}^{m \times n}$ matrix, where $m \ge n$, can be decomposed (QR factorization): $$A = \begin{bmatrix}Q_1 | Q_2 \end{bmatrix}\begin{bmatrix}R\\0\end{bmatrix}$$ where $Q_1$ and $Q_2$ has orthonormal columns, and $R$ is upper triangular.

I'm implementing a routine (based on Householder reflections) which calculates $Q_1$ and $R$ (so called thin/reduced QR decomposition).

My question is: is it possible to calculate $Q_1$ without calculating $Q_2$? The problem is that a Householder matrix is $\mathbb{R}^{m \times m}$, and $Q_1 \in \mathbb{R}^{m \times n}$, so I cannot multiply them. My routine currently calculates $Q=[Q_1|Q_2]$, and then throws away the $Q_2$ part.