We investigate the asymptotic behavior of the cosmological field equations in Symmetric Teleparallel General Relativity, where a nonlinear function of the boundary term is introduced instead of the cosmological constant to describe the acceleration phase of the universe. Our analysis reveals constraints on the free parameters necessary for the existence of an attractor that accurately represents acceleration. However, we also identify asymptotic solutions depicting Big Rip and Big Crunch singularities. To avoid these solutions, we must impose constraints on the phase-space, requiring specific initial conditions.