It has been conjectured that all graded Artinian Gorenstein algebras of codimension three have the weak Lefschetz property over a field of characteristic zero. In this paper, we study the weak Lefschetz property of associated graded algebras A of the Apéry set of M-pure symmetric numerical semigroups generated by four natural numbers. In 2010, Bryant proved that these algebras are graded Artinian Gorenstein algebras of codimension three. In a recent article, Guerrieri showed that if A is not a complete intersection, then A is of form A=R/I with R=K[x,y,z] andI=(xa,yb−xb−γzγ,zc,xa−b+γyb−β,yb−βzc−γ), where 1≤β≤b−1,max{1,b−a+1}≤γ≤min{b−1,c−1} and a≥c≥2. We prove that A has the weak Lefschetz property in the following cases:•max{1,b−a+c−1}≤β≤b−1 and γ≥⌊β−a+b+c−22⌋;•a≤2b−c and |a−b|+c−1≤β≤b−1;•one of a,b,c is at most five.