If there is a plank length, mass, and time doesn’t that mean that their is a quantum of acceleration? and if that is true then doesn’t that mean you couldn’t instantly accelerate an object to any speed below c instantly? even though it would use a monstrous amount of energy to accelerate something to near light speed

The existence of a Planck length does not imply that there is a quantum of length. Also, the quantization of certain variables is not to be confused with their uncertainty. For example, the momentum of a particle in infinite space is not quantized, yet the momentum and position can not be determined simultaneously with arbitrary precision. The three components of angular momentum can not be simultaneously determined either, but angular momentum is always quantized.

Nope. There is no quantum of acceleration. What is quantized in quantum mechanical systems are neither position nor momentum (both operators have a continuous spectrum for free particles) but a quantity called “action”. The basic unit of quantum mechanics, Planck’s constant h, has physical units [Js]. Quantization makes sure the action in a system changes by units of h. This does not mean any physical coordinates, momenta or derived quantities like classical acceleration are quantized.

“and if that is true then doesn’t that mean you couldn’t instantly accelerate an object to any speed below c”

We know of no inherent limit to accelerate objects fast. Indeed, the classical accelerations of particles interacting at high energies would be absolutely enormous. In a typical particle reaction an energy of 10^4 GeV can be lost over a distance of 10^-15m or less. I leave it to you to calculate the resulting acceleration both in the classical as well as relativistic framework. And there are absolutely no visible effects of this “semi-classical” term on the behavior of the system. All interactions can be written in terms of position and momentum operators and quantum field theory yield perfect results within the limits of our measurement precision. Of course, for charged particles “acceleration” results in Bremsstrahlung, but quantum electrodynamics gets that right without using the term “acceleration” even once.

These two “answers” do not address the question. Of course, in non-relativistic Quantum Mechanics acceleration is not quantized (answer 2). However, if gravity is to be quantized (to be sure, nobody has succeeded doing this), and if the Equivalence Principle holds, then acceleration has to be quantized as well.