FEniCSx uses iso-parametric elements, meaning that you can describe the cell with a P-th order Lagrange element (usually with equispaced dofs).
The quadrature scheme is chosen depending on the form (you can also set these yourself, see: Integration of matrix elements - #2 by dokken and its referenced) and the integrals are performed by mapping the integral to the reference element.
If you want to go into how the code actually looks like, you can formulate everything with ufl and Basix, as done in: dolfinx/cpp/demo/poisson/poisson.py at main · FEniCS/dolfinx · GitHub
and call python3 -m ffcx poisson.py to generate a header file with C code for assembling the local tensor of a and L