I'm not a Quantum Espresso expert, but I'll try to answer this as best I can. There are two aspects to this answer: 1) whether you have a GPU-enabled version of Quantum Espresso; and 2) whether you should use your GPUs anyway.
GPU port of Quantum Espresso
Using GPUs effectively requires substantial changes to the source code (e.g. OpenACC, OpenMP5, CUDA-Fortran), and so you need to make sure you are using a version of Quantum Espresso which has those changes. You will also need a compiler which understands the GPU additions, which probably means the NVIDIA compiler.
The previous GPU project was led by Filippo Spiga and was very quick, but made heavy use of CUDA-Fortran (only supported by NVIDIA) and I think it evolved into a separate branch, and it and the "main" Quantum Espresso diverged significantly.
There is a new GPU port of Quantum Espresso which is intended to stay "in sync" with the main developments. This has been released, but I'm not sure what state it's in -- it may still be a test release.
Regardless, you will need the NVIDIA Fortran compiler and the CUDA toolkit (both are free). You will not be able to use Intel's compilers and I doubt that you can use GNU's either.
GPUs for DFT simulations
The GPUs which people usually use for DFT simulations are not graphics cards, they are dedicated compute-platforms (GPGPUs) which use technology based on graphics cards.
The "built-in" Intel GPU is unlikely to be useful; it has relatively low performance, it probably has no fast, dedicated RAM, and it is unlikely to be supported by your compiler. It is also very difficult to program for two different kinds of GPUs in the same system, and I doubt either QE or the compiler would cope.
The NVIDIA GPU probably has a good headline performance, and some fast, dedicated RAM, but I'm confident that it will only support up to single-precision arithmetic in hardware. Accurate DFT simulations rely on double-precision, and whilst the cards can emulate this in software it is extremely slow.
Much of the speed of a GPGPU is only attainable at all because they also have dedicated, high bandwidth memory (e.g. HBM2), which may not be true of a laptop model. You typically want a lot of dedicated RAM as well, which most consumer cards do not have.
If we look at the NVIDIA GPU-based offerings, you end up looking at a Titan or Tesla card before you have enough fast RAM and any significant double-precision performance for DFT simulations. Both of these have error-corrected (ECC) RAM, which I also consider important to getting reliable, publishable results (this goes for CPU calculations too, of course).