Is the G-constant in Newtons equation the same as "G-force"

People often say "g-force" to mean the force that a person feels when they are accelerating, such as a centrifugal force, that they just measure in units of 'g' where each unit of force is the equivalent of the same gravitational force that the person would feel (on the Earth's surface).

For instance, if a person normally experiences a gravitational force of 30 Newtons, then if when they are taking off in an aircraft they feel a force of 60 Newtons pushing them in their seat, then they say "I felt two g's."

I don't hear things like "g-force" in physics class, I think it's primarily used when discussing forces acting on people's bodies.

The G-constant in Newton's law of gravitation is just a number, like pi or e (except it has dimensions, unlike pi or e). Newton's law of gravitation says that a body of mass m1 located a distance r from a body of mass m2 feels a gravitational force of....

F=G*m1*m2/r^2

The constant G is what makes that formula work. G is a property of the universe, it has to be measured and it's the same everywhere. If the first mass m1 happens to be the Earth's mass (call it M), and if the distance r happens to be the average distance between a person and the Earth's center of mass (call it R), then we get tired of writing:

F=G*M*m2/R^2

And instead write

F=m2*g

Where we just say g=GM/R^2. Saying F=mg is much shorter and is pretty accurate for anything near the surface of the earth. This g is an acceleration and is darn close to 9.8 m/s^2 or 32 ft/s^2, so if I feel two g's pushing me back in my seat, I feel 18.6 m/s^2 of acceleration.

No. G in the equation is a constant. The normal force of gravity, also called weight, is W = mg = mGM/r^2, where g = 9.81 N/kg is the normal gravity field value on the surface of the Earth. And that g is called 1-g. As you can see, the other G, goes into making up the value of g.