How to convert/rotate Euler angles from one coordinate system to another

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I have two Euler angles (Z and Y'). They are made for a coordinate system where X is forward, Z is up, and Y is to the left.

I have another coordinate system that is rotated around X 180 degrees. So X is forward, Z is down, Y is to the right.

I need to figure out how to convert my original Euler angles into ones for this new coordinate system.

Please excuse any odd phrasing - I am not a mathematician and not very familiar with Euler angles or transformations/rotations in general.

This is for C++ software, so any familiarity with boost's qvm library is appreciated.

I think I can construct a rotation matrix for a 180 degree rotation around X, but at that point I become lost. Rotation matrices are meant to be used with vectors, not Euler angles, aren't they? And I can't find anything on converting between the two.