Beilstein doesn't give a ref for 1,4-dimethoxybenzene chloromethylation. Seems like this doesn't work.
But 1,4-diomethoxybenzene chloromethylation is described, producing 1,4-dimethoxy-2,5-bis-chloromethyl-benzene:
H3PO4, Hg(!!!), aq. HCl; Zh. Obshch. Khim. 29, 1959, 3746; engl. ed. 3704, 3705
or
aq. HCl, HCl, acetic acid; Bull. Chem. Soc. Jpn, 48, 1975, 497-504
2,5-dimethoxytoluene chloromethylation in concentrated HCl:
Helv. Chim. Acta 39 (1956), 1564, 1575
2-Bromo-1,4-dimethoxybenzene chloromethylation:
Bull. Soc. Chim. Fr. 1953; C46
1,4-dimethoxynaphtalene chloromethylation:
64%, gas. HCl, conc. HCl, dioxane, H2O, 20 min, 55-60°C;
J. Org. Chem 48, 19, 1983, 3265-68
Another interesting ref with lots of chloromethylations in it:
Chem. Ber. 80, 1947, 391-399