Yes. You can represent the coordinates of an object in various reference frames from using transformation matrices. Here, I assume the origins of the reference frames are aligned based on your description, so you only need to use a rotation matrix.
You can compute the rotation matrix for rotating reference frame $A$ on reference frame $B$ by rotations about each axis in sequences. The rotations around $x$, $y$, and $z$ axes are given as follows:
$$
R_x(\theta) = \begin{bmatrix} 1 & 0 & 0\\ 0 & \cos{\theta} & -\sin{\theta} \\ 0 & \sin{\theta} & \cos{\theta} \end{bmatrix}
$$
$$
R_y(\phi) = \begin{bmatrix} \cos{\phi} & 0 & \sin{\phi} \\ 0 & 1 & 0 \\ -\sin{\phi} & 0 & \cos{\phi} \end{bmatrix}
$$
$$
R_z(\psi) = \begin{bmatrix} \cos{\psi} & -\sin{\psi} & 0\\ \sin{\psi} & \cos{\psi} & 0 \\ 0 & 0 & 1 \end{bmatrix}
$$
Now, the order that you apply the rotation matters, so if you want to rotate around $z$, then $y$, then $x$, the rotation matrix is computed as follows:
$$
R = R_x(\theta)R_y(\phi)R_z(\psi)
$$
Now, if the angles represent the amount that you rotate frame $A$ to align frame $B$, then the above rotation matrix can be used to transform points from frame $B$ to frame $A$, which we can denote by ${}^A R_B$. This notation represent the orientation of frame $B$ in reference frame $A$. If we want to transform points from frame $B$ to frame $A$ ,then we take the inverse of the rotation matrix ${}^B R_A = {}^A R_B^{-1}$.
Now, consider a point on the rigid body of an object in frame $A$ denoted by ${}^Ap$. To get the position of that point in frame $B$, you simple multiply by the appropriate rotation matrix as follows:
$$
{}^Bp = {}^B R_A {}^Ap
$$
where ${}^Bp$ is the same point as ${}^Ap$ but represented in frame $B$. You can repeat this process for any point on rigid objects.
