does the body release glucagon when under stress?

Answer 1

yes, that's why certain situations of stress can lead to a misdiagnose of high bllod sugar. The body releases cathecolamines, glucagon and other hormones that inhibit insulin which is the one responsible for maintaining normal levels of glucose or sugar in the blood.

Answer 2

The Pancreas regulates the body's blood sugar levels by producing and releasing two hormones: insulin and glucagon. Insulin is released to lower blood sugar levels by signalling that your body convert it to fat, and store it in your fat cells. Glucagon is released when blood sugar levels drop, stimulating the breakdown of stored fat into sugars which are released into the bloodstream.
So yes, under stress the pancreas will release both.