Yes. I had one during my first pregnancy and no issues. Make sure the salon has good ventilation is the key.
Not sure what a perm entails or how long something like that would take but when I had my 12 week check up I had been told I could still get my hair colored and nails done if I wanted to, I think as long as you're in a ventilated area it's fine also with all the hormones my hair has become difficult to straighten myself so not sure if a perm would come out as it normally would ...hope this helps!