Can a magnet damage a compass?

Yes a magnet can damage a compass.

The compass needle is a ferromagnetic material.

The degree to which a ferromagnetic material can "withstand an external magnetic field without becoming demagnetized" is referred to as its coercivity.

Another magnet near the compass needle imposes a magnetic field upon the compass needle. It is a matter of the strength of the magnetic field imposed upon the compass needle and the coercivity of the needle material whether or not the magnetic properties of the compass needle are damaged.