You can use stderror:
julia> using GLM
julia> x = rand(10); y = rand(10);
julia> res = lm(@formula(y~x), (;x, y))
StatsModels.TableRegressionModel{LinearModel{GLM.LmResp{Vector{Float64}}, GLM.DensePredChol{Float64, LinearAlgebra.CholeskyPivoted{Float64, Matrix{Float64}, Vector{Int64}}}}, Matrix{Float64}}
y ~ 1 + x
Coefficients:
─────────────────────────────────────────────────────────────────────────
Coef. Std. Error t Pr(>|t|) Lower 95% Upper 95%
─────────────────────────────────────────────────────────────────────────
(Intercept) 0.768012 0.0978733 7.85 <1e-04 0.542315 0.993708
x -0.455953 0.218396 -2.09 0.0703 -0.959576 0.0476696
─────────────────────────────────────────────────────────────────────────
julia> stderror(res)
2-element Vector{Float64}:
0.09787333346669594
0.21839629429830254
Edit: see here for more access methods: Examples · GLM