(A) Can you tell me how to apply for [a] family allowance for my children? (B) Can you tell me how I can apply for [a] family allowance for my children? To native speakers, both questions (A) and (B) are asking the same thing. The slight difference is that you are expressing an intention to do something with a to-infinitive in (A) while you are seeking advice on how to enable yourself do something with the modal verb "can" in (B).

Yes, both are OK and both mean the same thing. But both would be better with “a” or “the” in front of “family allowance.” It does not sound natural without an article.