Particles in physics in current terminology is "elemenntary particles", which are the building blocks that form atoms molecules and radiation that we observe macroscopically in bulk. These can be created and destroyed during the processes of stellar evolution, and particularly photons, which are bosons, have no limit on their number at all. There is no limit to the number of particles currently or at any time after the Big Bang.
If you mean if "the number of atoms is finite" a tentative number could be estimated from the mass of galaxies, the number of galaxies in galactic clusters and the number of clusters in a homogeneous universe ( ignoring dark matter since we know next to nothing about it) One could come out with a number by dividing the average galaxy mass by the average atomic weight estimated for the distribution of atoms in a galaxy.This number would be just an estimate and would not be constant because stars evolve, sometimes becoming black holes, sometimes atoms fuse into heavier atoms, sometimes atomic nuclei decay to two or more fragments etc.
By both definitions of particles , their numbers are not constant.