Both are technically correct. Here's a review that goes into some detail about the process: IgG Placental Transfer in Healthy and Pathological Pregnancies. It's worth a read if you're interested in the subject. In particular, it cites a study of immunoglobulin transport which:
demonstrated a continuous rise in IgG levels in the fetal circulation between 17 and 41 weeks of gestation. Fetal IgG concentrations were only 5%–10% of the maternal levels at weeks 17–22 but reached 50% of the maternal concentrations at weeks 28–32. The majority of IgG is acquired by the fetus during the last 4 weeks of pregnancy, and fetal IgG concentrations usually exceed maternal ones by 20%–30% at full term.
So it begins a wee bit before week 17, so definitely within the second trimester, but it is only by the third that concentrations begin to be truly meaningful. However, most of the antibody acquired, going above the maternal level, is done in the last month.