  In this work, we consider an extension of the symmetric teleparallel
equivalent of General Relativity (STEGR), namely, $f(\mathbb{Q})$ gravity, by
including a boundary term $\mathbb{B}_Q$, where $\mathbb{Q}$ is the
non-metricity scalar. More specifically, we explore static and spherically
symmetric black hole and regular black hole solutions in
$f(\mathbb{Q},\mathbb{B}_Q)$ gravity coupled to nonlinear electrodynamics
(NLED). In particular, to obtain black hole solutions, and in order to ensure
that our solutions preserve Lorentz symmetry, we assume the following relation
$f_Q = -f_B$, where $f_{Q}=\partial f/\partial\mathbb{Q}$ and $f_{B}= \partial
f/\partial\mathbb{B}_Q$. We develop three models of black holes, and as the
starting point for each case we consider the non-metricity scalar or the
boundary term in such a way to obtain the metric functions $A(r)$.
Additionally, we are able to express matter through analytical solutions for
specific NLED Lagrangians ${\cal L}_{\rm NLED}(F)$. Furthermore, we also obtain
generalized solutions of the Bardeen and Culetu types of regular black holes,
by imposing specific metric functions.
