Missed prayers during the forbidden times
Broad division of Prayers
Prayers can broadly be divided into obligatory prayers and non-obligatory prayers. You pray obligatory prayers at the time Allaah has specifically stipulated for those prayers.
You can pray non-obligatory prayers at times when you please. The exception are the Sunnah prayers which follow or precede the obligatory prayers. These are non-obligatory prayers. But you have to pray them either before or after the obligatory prayer they are related to.
Generally speaking if you miss a non-obligatory prayer or a Sunnah prayer you do not have to make it up.
You have to make up an obligatory prayer you miss. For example, you may oversleep or forget to pray an obligatory prayer. If this happens, you now have to make it up.
Forbidden times to pray
There are times when the Messenger of Allaah (sallallaahu alayhi wa sallam) forbade Muslims to pray.
According to a large group of Muslim scholars, with some claiming it to be the majority of scholars, you can pray a obligatory prayer you have missed during these forbidden times. Obligatory prayers you miss are an exception to the rule that you cannot pray during the forbidden times because of the following hadith of the Messenger of Allaah (sallallaahu alayhi wa sallam):
مَنْ نَامَ عَنْ صَلاةً أَوْ نَسِيَهَا فَلْيُصَلِّهَا إِذَا ذَكَرَهَا
He who sleeps [and] misses a prayer or forgets [to pray] it then let him pray it when he remembers it.
The phrase when he remembers it includes any and all times, including the forbidden times. This is the opinion of the following scholars of Islaam:
أبو العالية
النخعي
الشعبي
الحكم
حماد
مالك
الأوزاعي
الشافعي
إسحاق
أبو ثور
ابن المنذر
The opinion has also been transmitted from Ali (Allaah is pleased with him) and other Companions.
You can read Ibn al-Qudaama's book al-Mughni to confirm the above. The exact page will differ based on the publisher of the book.
Alternatively, you could download al-Maktabah al-Shaamilah and turn to Volume 2 Page 81 of the digital version of the book to find the above information in it.
Only the Ahl al-Ra'y differ and do not allow you to pray an obligatory prayer you miss during the forbidden times.
If you are a Hanafi and strictly adhere to your Madhab, then the above information would not be applicable to you because you would not pray during the forbidden times even if it meant making up a missed obligatory prayer.