Hydroquinone can be formylated directly by addition in toluene or benzene of SnCl4 or FeCl3 and tributylamine.  Solution is heated for 8-24 hours, poured into acidified water, extracted with ether, and purified via steam distillation.  

JCS Perkins I, (1980) 1862-1865

Probably your most practical formylation is the method of Reimer and Teimann, which employs only NaOH, chloroform, and your phenol as reagents.  However, the yields are usually low, and in the case of hydroquinone, the reaction appears not to work.  However, formylation of 4-methoxyphenol proceeds smoothly, with reported yields ranging from 50%-70%.  So you could methylate hydroquinone using dimethyl sulfate, formylate it, and then demethylate.

Chemical Reviews, Wynberg H., 60, 1960

Would methyl iodide work as a methylating agent for hydroquinone?

You also need formaldehyde, of course (or paraformaldehyde).

Oh yeah.  Forgot to add that part. *bonks self on head*

It might be difficult to prevent over-methylation with either dimethyl sulfate or methyl iodide.

Perhaps dropwise addition of (half the molar amount of) methylating agent to a vigorously stirred solution of phenol would do it.  

What solvent and other reaction conditions are needed when using methyl iodide?

hydroquinone can be converted into p-methoxy benzene:
MeOH solvent
2.1 eq KOH
2.2 CH3I
16h / rt

during work-up, wash mixture with 15% NaOH solution to remove unreacted hydroquinone.

p-methoxy anisole or p-methoxy phenol?

p-methoxy anisole