Indirect questions follow the same tense-shift rules as ordinary indirect speech. If
la concordance des temps isn't familiar yet, it's worth a look at
reporting speech in French first, since this page builds directly on it. What's new here is what happens to the question itself.
A direct question has a question mark and often an inversion or est-ce que. Once it's reported indirectly, all of that disappears, the question mark, the inversion, and, depending on the type of question, the question word itself changes shape.
Yes/no questions take si
A direct question that can be answered with oui or non, whether it uses est-ce que, inversion, or just rising intonation, becomes si once it's reported.
Question posée…
« Vous êtes allés au concert de Gojira en juin ? »
est-ce que / inversion → si
êtes allés→étions allés
Il m’a demandé si nous étions allés au concert de Gojira en juin.
No question mark, no inversion, and the passé composé becomes a plus-que-parfait, since a demandé is in the past.
Interrogative adverbs stay the same
Questions built with où, quand, pourquoi, comment or combien keep that same word once reported. Two things change around it: any inversion disappears, so the subject sits in front of the verb the way it would in an ordinary statement, and the tense shifts according to the same concordance des temps rule.
Question posée…
« Pourquoi tu ne m’as pas prévenue plus tôt ? »
pourquoi — no change
m’as prévenue→l’avais prévenue
Elle a demandé pourquoi je ne l’avais pas prévenue plus tôt.
Pourquoi carries straight over. The pronoun also shifts, tu/m’ become je/l’, since someone else is now reporting the question.
Question posée…
« Comment a-t-il pu s’échapper ? »
comment — no change
a pu→avait pu
Ils se sont demandé comment il avait pu s’échapper.
Notice the word order too: no more inversion (a-t-il becomes simply il avait).
Qu’est-ce qui and qu’est-ce que transform
These two both mean roughly "what," but they ask about different things, and each turns into something different once reported.
Qu’est-ce qui is used when the unknown thing is doing the action, it's the subject of the question. "What happened?", you don't know what caused the event, that "what" is the one performing the action. This becomes ce qui.
Qu’est-ce que is used when the unknown thing is having the action done to it, it's the object of the question. "What do you think?", the "what" is the thing being thought about, not the one doing anything. This becomes ce que.
Question posée…
« Qu’est-ce qui lui est passé par la tête ? »
qu’est-ce qui→ce qui
est passé→était passé
Elle a demandé ce qui lui était passé par la tête.
Qu’est-ce qui asks about an unknown subject, and becomes ce qui.
Question posée…
« Qu’est-ce que tu penses de ce tableau ? »
qu’est-ce que→ce que
tu→je
Il m’a demandé ce que je pensais de ce tableau.
Qu’est-ce que asks about an unknown object, and becomes ce que. The verb also shifts, présent to imparfait, following the usual rule.
Both qu’est-ce qui and qu’est-ce que have a shorter, more formal equivalent: que with inversion. Qu’est-ce qui lui est passé par la tête ? can also be written Que lui est-il passé par la tête ?, and Qu’est-ce que tu penses de ce tableau ? can also be written Que penses-tu de ce tableau ?. Either way, the indirect result is exactly the same, ce qui or ce que.
Two more things carry over from any indirect question: the word order stays like a normal statement (subject before verb, no inversion), and there's never a question mark.