Can I use a reed relay instead of a reed switch with an Arduino to detect when a magnet is close?
Simplest answer. Try it. The relay is an electromagnet. It depends on magnetism to work. It might be shielded, it might not. Use the continuity feature on a multimeter and a magnet and test it.

If it works, you would wire it like you would a regular reed switch, with the L2 pins connected to ground and an Arduino digital pin set to input (with pullup).