Technically speaking, increasing ISO does not actually "increase noise". When you increase the ISO setting, you are actually just instructing the camera to change what level of charge represents "maximum saturation" (the point at which a pixel should reach its maximum numeric value, which in the case of a 14-bit sensor is 16384.)
The primary cause of noise in an image is the random distribution of photons as they reach the sensor. The fewer photons reaching the sensor, the more apparent their random distribution will be...thus the visible "grain". The amount of noise you see in a photo is inversely related to the amount of light reaching the sensor. Less light, more noise...more light, less noise. For a given exposure time, when you get less light down the lens, you expose fewer photons per pixel, and since photons are effectively distributed randomly, some pixels may get less than they should while others get more than they should. That uneven distribution is the specific cause of photon shot noise. When you get more light onto the sensor, the random nature of light will distribute more and more evenly across the area of the sensor, and as the average number of photons per pixel increases and normalizes, the effects of photon shot noise will lessen.
When you crank up the ISO to produce a "proper" exposure for that small amount of light, you amplify the sensor signal, which exaggerates the effect that photon shot noise has. Alternatively (assuming an ideal sensor that introduces no electronic noise of its own), you could always expose at ISO 100, then boost exposure by the necessary number of stops in post, and experience the same effect. It is not the amplification, be it analog or digital, that creates noise...it simply enhances the effects of noise.
- "My question is, will the image noise of a high ISO/fast shutter speed be the same as a low ISO/slow shutter speed where both settings provide the same total amount of exposure?"
Shutter speed is something you can use to control what ISO setting you need. Shutter speed is exposure time, and if you can only get 1/4 the amount of light down the lens that you need to be able to use ISO 100, you can always expose four times as long. There are caveats to this...if you are photographing anything with motion, you'll see motion blur in the final photo if shutter speed is too slow. Slower shutter speed can also lead to blur from camera shake, which can reduce the sharpness and contrast of your photo. If you are photographing a still scene...such as a landscape or still life, you probably have the option of reducing shutter speed rather than increasing ISO to achieve a proper exposure. The longer exposure will allow more light to reach the sensor, and the use of a lower ISO will require less amplification, reducing noise on two fronts (more light/less noise, less amplification/less noise.)
So the answer to your question is NO, if you trade a high ISO/fast shutter for low ISO/slow shutter, you should end up with LESS NOISE (but potentially more motion blur or blur from camera shake if you are shooting hand-held.)