Firstly, we shall note that Shia’s sources of religion knowing is consists of 3 things: Quran/Sunnah (speech/practice/statement of the Prophets and Shia Imams)/and Aql (intellect), Secondly it should be paid heed that there is not all details of Ahkam (religious orders/rulings) in the Quran.
Anyhow, concerning the query: Yes, women have to prayer (offer their Salah -prayer-) with hijab even when they are alone at home according to Shia’s view. And as much as I know, all of Shia scholars/Maraja’ (as a kind of source) declare so, and actually doesn’t allow to offer Salah without hijab (such as Ayatollah Khamenei, Ayatollah Sistani, Ayatollah Makarem, etc.)
Based on Shia’s related explanation, while a person see himself (herself) in front of Allah during Ibadah (worship), so it is good to wear his/her best clothes. Hence, e.g. the best covering for women is complete covering which is the symbol of Efah (chastity), although there is no non-Mahram there. As a related source: Porseman, Vol.6 Pg.151.
E.g., Imam Muhammad al-Baqir (a.s.) -as a related source- mentioned:
The woman must prayer in a Lebaas (clothes) which covers her. Source:
(Wasaael-al-Shi’a, vol.4, pg.405/406).
So, I assume e.g. Shia might infer/conclude (as a conclusion) from this hadith (and other similar hadiths) that: It is completely a proven issue for Shia (and I think Sunni) that it is Wajib for women –based on Qur’an/hadiths- that they must be with hijab in front of non-mahram(s). But in the above-mentioned canonical-narration it doesn’t speak regarding the issue of hijab in front non-mahram, whereas the gist of the subject explicitly is mentioning regarding the significance/necessity of being covered in prayer (without pointing out about the attendance/not attendance of non-mahram…). Hence, easily/logically/fairly it can be concluded that it is illustrating regarding the necessity of being covered during the prayer in general (in every situation), and factually it doesn’t limit being with hijab just in attendance of non-mahram.
And other related hadiths …
Sources: http://www.hawzah.net & http://www.askquran.ir
Updated (and extra helpful information):
According to Imam Khomeini as a Shia scholar/Marja’al-taqlid,
woman must cover all of her body -even head and hair-, (but) it is
not necessary to cover the face (the amount of the face which is
washed in Wudhu) and hands till the ankle(s). Meanwhile, if I mistake
not, it is the Fatwa of almost all Shia scholars/Marja’al-Taqlids.
1. امام خمینى، توضیح المسائل، محقق، مصحح، قلىپور گیلانى، مسلم، ص
177، م 771، چاپ اول، 1426ق.
Further accurate issues: this and
that.
And finally as a significant helpful hadith (particularly for the questioner), Imam Baqir -a.s.- said regarding the Lebaas of woman during Salah (prayer):
The woman, must say prayer in Der’ and Maqna’ah (but by this condition
that the Der’ must be thin).
Ref.: (al-Kulayni , al-Kafi …)
. کلینی، محمد بن یعقوب، الکافی، محقق و مصحح: غفاری، علی اکبر،
آخوندی، محمد، ج 3، ص 394، دار الکتب الإسلامیة، تهران، چاپ چهارم،
1407ق.
Definition of Der’ and Maqna’ (that hopefully I define them accurately in English based on the extant non-English/origin sources):
Der’: in Fiqhi terminology, Der’ is said/called to a kind of woman dress which is with sleeves …
Maqna’ah: this is Khemaar, and Khemaar is the thing by which woman covers her head and hairs…
For further helpful/detailed information (and likewise the source), you can refer to the following link (which unfortunately is only in Farsi (Persian)) (at least I provided most of my answer based on it):
