For the family of generalized $q-$Onsager algebras associated with simply-laced affine Lie algebras ${\widehat g}$, new relations between certain monomials of the fundamental generators are proposed. These relations can be seen as deformed analogues of Lusztig's higher order $q-$Serre relations associated with ${\cal U}_q({\widehat g})$, which are recovered as special cases in relation with certain coideal subalgebras of ${\cal U}_q({\widehat g})$.