Newtonian fluids exhibit no resistance to deviatoric stresses and can deform indefinitely. As a result, most researchers adopt the Eulerian description, modeling the fluid flow within a fixed spatial domain. However, this approach is not directly applicable to moving boundary flows, such as those involving multiple fluid flows or fluid-structure interactions. The challenge becomes even greater when topological changes in the fluid domain are present. An alternative for addressing such problems is the Particle Finite Element Method (PFEM), which combines finite elements with particle concepts. In this approach, the fluid is represented by a cloud of particles upon which a finite element mesh is constructed to solve the motion equation within a Lagrangian framework. This mesh is continuously destroyed and rebuilt. Despite its advantages, applying boundary conditions in a Lagrangian framework for spatially fixed inflow, outflow, or even slip wall boundaries is challenging, as the boundary moves with the particles, deforming the analysis domain. In this work, we present a particle position-based space-time finite element formulation for incompressible Newtonian fluid flows, based on [1] capable of handling spatially fixed boundaries with inflow and outflow or slip wall conditions. The method employs mixed position-pressure finite elements with Petrov-Galerkin pressure stabilization. The space-time discretization is structured in the time direction, with space-time shape functions defined as a tensor product of linear spatial shape functions and quadratic temporal shape functions. This structure allows the space-time domain to be divided into slabs, similar to [2], which are solved progressively, with velocities and positions from the previous slab serving as initial conditions for the current one. To handle spatially fixed boundaries, we propose a particle relocation approach that aims to maintain the PFEM mesh quality while ensuring fixed inflow, outflow, or slip wall conditions. The formulation is verified through numerical examples, demonstrating its robustness and efficiency in simulating incompressible flows in closed domains, such as Poiseuille and Hagen-Poiseuille flows.