Should I be worried about a surge protector near my pillow
It is very doubtful that you are in danger as a result of electromagnetic radiation. The primary hazard from a surge suppressor would be that it may conduct large currents in a lightning strike, possibly even blowing up, and that it may become very hot in the case of certain types of internal failures. It could even start a house fire. Moving it farther away from your bed couldn't hurt. If possible, mount it on a surface with at least some amount of fire resistance, such as plaster, drywall, or on a floor made of masonry or tile. Do not allow it to be covered with flammable materials or anything that might prevent it from dissipating heat effectively (such as a blanket or clothing).

It would be a good idea to make sure the suppressor has the correct safety agency rating applicable for where you live. That would be the UL in the USA. Properly designed surge suppressors should have some type of thermal disconnect or thermal fuse to make sure that they do not continue to operate when they get too hot.

But shoddy devices, or old ones may be missing this safety feature.

It is a good idea to have a smoke detector in every bedroom, too.