I have a Smith and Wesson Mod. 37 in cal. .38. Could I use + P ammo in the pistol?

Could you, yes. The real question is should you. The +P round will increase the wear on your pistol and may or may not provide the extra benefit you desire.

Personally, I would use the serial number, investigate the age of the weapon, then follow company directions. The J frame Smith has been a very successful firearm, manufactured in a variety of models since 1950, so obviously some of these pistols would not have the magnum frame and will not be certified +P. Smith makes a great revolver and the individual pistol will probably tolerate the occasional use of +P ammo. The continual use of said ammo on a regular ongoing basis (read competition shooting) in a non +P rated weapon would obviously lead to complications.

What little information I found indicates the more modern (manufactured since 1990) will be able to withstand the +P ammo, but those weapons manufactured prior to this period may have issues with this option. Given that, the use of a non +P defensive round such as the Federal's 110 grain .38 Special HydraShoks (PD38HS3H) might be a better option.

I own a Chief's special, model 60. Using +P in this weapon makes for reduced precision and accuracy due to the recoil. I like the HydraShoks just fine (well, except the expense).