Key Point: Resonance will only occur when it stabilizes a molecule.
Just because resonance can occur does not mean that resonance will occur. Consider the case of cyclobutadiene, resonance interaction would produce a square molecule that is destabilized (antiaromatic). Therefore the molecule adopts a rectangular geometry to avoid the destabilizing resonance interaction.
I've drawn two of the resonance structures for the ionized form of p-methoxyphenol below. Resonance would destabilize the ionized form because of resonance structures like A where the electron donating resonance effect of the methoxy group places two negative charges close to one another. Rather than destabilize the molecule, the methoxy group simply favors a conformation such that the oxygen lone pairs are twisted out of the plane of the aromatic p-orbitals and cannot overlap with them most of the time.
On the other hand, resonance structures like B are stabilized by the electron withdrawing inductive effect of the methoxy group. If we examine resonance structure C, the counterpart of resonance structure B for the p-methylphenol analogue, we see that since the methyl group is inductively electron releasing, resonance structure C is destabilized.
Conclusion: Since both the methoxy and methyl group release electrons through resonance and since this would destabilize the ionized, anionic phenols, resonance is minimized through conformational effects. Therefore, the smaller inductive effects of the two substituents control the situation with the methoxy stabilizing and the methyl destabilizing the ionized form. Hence, p-methoxyphenol is slightly more acidic than p-methylphenol.