This is just to emphasize and support the answer given by @UmH and hopefully give some additional information.
First of all, before addressing the question, I have to note that you have picked out two verses about the sacred months, but you have already skipped the first mention in the Qur'an, even if it would fit the topic exactly, since even here the singular was used.
Allah the Almighty says in (2:217):
They ask you about the sacred month - about fighting therein. ...
Note that this is the topic that is later in the qur'an addressed again in surat at-Tawba (#9) which is usually also referred to by Baraa' براءة. And this verse has a relation to both verses you've addressed as I will try to explain.
Secondly in Arabic language it is not unusual to express plural be it the general plural or dual (for two persons) by singular and vice versa so the genus is interchangeable see for example: What happend to the second garden of the disbeliever? or Examples of verses of the Qur'an with two different readings? to give a couple of examples.
Now back to your both verses:
The general agreed upon understanding: month stands for the genus so it means all sacred month
We read in the tafsirs on (5:2) the following statements of scholars:
In the following I'll be translating from Arabic, since these translations are of my own -with the support of translation tools-, take them carefully!
Ibn al-'Aarbi said in his Ahkam al-Quran (source):
قوله تعالى : { ولا الشهر الحرام } : قد بينا في كل مصنف أن الألف واللام تأتي للعهد وتأتي للجنس ; فهذه لام الجنس ، وهي أربعة أشهر يأتي بيانها مفصلة في سورة " براءة " إن شاء الله تعالى .
The Almighty’s saying: {Not (even)... the sacred month}: We have made it clear in every work that the alif and laam letters come to (express) the covenant and come to (express) the genus; Here this is the genus. And it is four months, which will be explained in detail in Surah “Baraa’,”(surat at-Tawba/9) Allah Almighty willing.
So the author is referring to the explanation in verse (9:36).
Similarly, you may find al-Qurtubi saying in his tafsir (source):
قوله تعالى : ولا الشهر الحرام اسم مفرد يدل على الجنس في جميع الأشهر الحرم وهي أربعة : واحد فرد وثلاثة سرد ، يأتي بيانها في " براءة " ; والمعنى : لا تستحلوها للقتال ولا للغارة ولا تبدلوها ; فإن استبدالها استحلال ، وذلك ما كانوا يفعلونه من النسيء
Allah the Almighty’s saying: “Not... the sacred month" is a singular noun that indicates the genus of all the sacred months, which are four: one singular and three consecutive, its explanation comes in “Bara’a”; The meaning is: Do not make it permissible for fighting or raiding, and do not change it. Replacing it makes it permissible, and that is what the Nasi'is used to do.
Again the verse (9:36) is mentioned as the explanation.
Sheikh Taher ibn 'Ashur states in his At-Tahrir wa at-Tanwir التحرير والتنوير (Source):
والشهر الحرام من الشعائر الزمانية ، والهدي والقلائد من الشعائر الذوات . فعطف الشهر الحرام والهدي وما بعدهما من شعائر الله عطف الجزئي على كليه للاهتمام به ، والمراد به جنس الشهر الحرام ، لأنه في سياق النفي ، أي الأشهر الحرم الأربعة التي في قوله تعالى منها أربعة حرم ذلك الدين القيم فلا تظلموا فيهن أنفسكم . فالتعريف تعريف الجنس ، وهو كالنكرة يستوي فيه المفرد والجمع .
The sacred month is one of the temporal rituals, and the sacrificial animals and garlanding are among the personal rituals. So the conjunction of the sacred month and the sacrifice and what comes after them among Allah's rituals is a conjunction of the partial with the whole of attention to it, and what is meant by it is the type of the sacred month, because it is in the context of negation, that is, the four sacred months of which, in the Almighty’s saying, “of these, four are sacred. That is the correct religion, so do not wrong yourselves during them.(9:36)” The definition is the definition of the genus, and it is like the indefinite article in which the singular and the plural are equal.
All the above explanations and many more that can be found in tafsirs (many exegetes didn't address the topic of genus and more explained the meaning of the sacred months) emphasize the overall truth of the statement that the sacred month here doesn't need to be specified since all four months have the same sacredness and same value. However, from a historical perspective, it could be of interest to know whether some single month was reported to be related to the verse in question.
Examples of singular reports stating that a specific month was meant
That's why we further read in at-Tahrir wa at-Tanwir (same source as above):
قال ابن عطية : الأظهر أنه أريد رجب خاصة ليشتد أمر تحريمه إذ كانت العرب غير مجمعة عليه ، فإنما خص بالنهي عن إحلاله إذ لم يكن جميع العرب يحرمونه ، فلذلك كان يعرف برجب مضر ; فلم تكن ربيعة ولا إياد ولا أنمار يحرمونه .
Ibn 'Atiyyah said: What appears to be the case is that Rajab was intended specifically so that the matter of prohibiting it would be more severe since the Arabs did not unanimously agree upon it. It was only specifically forbidding it as permissible since not all Arabs prohibited it, and for that reason it was known as Rajab Mudar(See in the sahih hadith for example here). Neither of the tribes Rabia, Iyad, and Anmar did consider it sacred.
وكان يقال له : شهر بني أمية أيضا ، لأن قريشا حرموه قبل جميع العرب فتبعتهم مضر كلها لقول عوف بن الأحوص :
وشهر بني أمية والـهـدايا إذا حبست مضرجها الدماء
وعلى هذا يكون التعريف للعهد فلا يعم . والأظهر أن التعريف للجنس ، كما قدمناه .
It was also called the month of the Umayyads, because the Quraysh forbade it before all the Arabs, so all of Mudar followed them, according to the words of 'Awf ibn Al-Ahwas:
And the month of the Umayyads and the sacrificial animals, when they are filled with blood
Based on this, the definition of the covenant is not general.
What is more apparent is that the definition is for the genus, as we have presented.
Note that Ibn 'Atiyyah himself also advocated for the genus explanation first in his own tafsir as you may read (Source):
وقوله تعالى : ولا الشهر الحرام ؛ اسم مفرد يدل على الجنس في جميع الأشهر الحرم؛ وهي كما قال النبي - عليه الصلاة والسلام -: "ذو القعدة؛ وذو الحجة؛ والمحرم؛ ورجب مضر؛ الذي بين جمادى؛ وشعبان"؛
And Allah the Almighty says: Not (even)... the sacred month. A singular noun that indicates gender in all the sacred months. It is as the Prophet - peace and blessings be upon him - said: “Dhul-Qa’dah; Dhul-Hijjah; Muharram; Rajab Mudar; which is between Jumada; and Sha’ban.”
وإنما أضيف إلى مضر لأنها كانت تختص بتحريمه؛ وتزيل فيه السلاح؛ وتنزع الأسنة من الرماح؛ وتسميه "منصل الأسنة"؛ وتسميه " الأصم "؛ من حيث كان لا يسمع فيه صوت سلاح؛ وكانت العرب مجمعة على "ذي القعدة؛ وذي الحجة؛ والمحرم"؛
But "Muddar" was added (mentioned beside it) because they were specific to prohibiting it. And remove the weapon; The tips are removed from the spears; They called it “the one who removes tongues.” They called it “the deaf”; From where he was, the sound of a weapon could not be heard; The Arabs agreed on “Dhul-Qa’dah, Dhul-Hijjah, and Muharram.”
وكانت تطول عليها الحرمة؛ وتمتنع من الغارات ثلاثة أشهر؛ فلذلك اتخذت النسيء؛ وهو أن يحل لها ذلك المتكلم "نعيم بن ثعلبة"؛ وغيره المحرم؛ ويحرم بدله صفرا؛ فنهى الله عن ذلك بهذه الآية؛ وبقوله: إنما النسيء زيادة في الكفر ؛ وجعل المحرم أول شهور السنة؛ ...
They were inviolable for a long time. refrain from raids for three months; that's why Al-Nasy' was invented; That is, it was declared permissible for them to speak (again)(refers to all kinds of weapon: meaning war is allowed) by the spokesman Na'ym ibn Tha’labah. And others for example the sacredness of al-Muharram; was lifted and replaced by declaring Safar as sacred; So Allah forbade that with this verse. And by his saying: Al-Nasy’i is only an increase in disbelief (See 9:37 an-Nasy': the postponing). HE further made Muharram the first month of the year. ...
After that the Qadi abu Muhammad ibn 'Atiyah quoted the above statement which ibn 'Ashur was referring to and added some further quotes and arguments:
قال أبو عبيدة : أراد رجبا؛ لأنه شهر كانت مشايخ قريش تعظمه؛ فنسبه إلى بني أمية؛ ذكر هذا الأخفش في "المفضليات"؛ وقد قال الطبري : المراد في هذه الآية رجب مضر.
Abu Ubaidah said: He emphasized Rajab; Because it is a month that the sheikhs of Quraish used to venerate; He attributed it to the Umayyads. This Al-Akhfash was mentioned in “Al-Mufadiyatiyat”; Al-Tabari said: What is meant in this verse is Rajab, Mudar.
قال القاضي أبو محمد - رحمه الله -: "فوجه هذا التخصيص هو - كما قد ذكرت - أن الله تعالى شدد أمر هذا الشهر؛ إذ كانت العرب غير مجمعة عليه"؛ وقال عكرمة : المراد في هذه الآية ذو القعدة؛ من حيث كان أولها؛ وقولنا فيها: "أول": تقريب وتجوز؛ إن الشهور دائرة؛ فالأول إنما يترتب بحسب نازلة؛ أو قرينة ما؛ مختصة بقوم.
Qadi abu Muhammad - may Allah have mercy on him - said: “The reason for this specification is - as I mentioned - that Allah the Almighty made the matter of this month strict, since the Arabs were not unanimously agreed upon it.” Ikrimah said: What is meant in this verse is Dhul-Qi’adah. From where it was first; Our statement about it: “first” means approximation and permissibility; The months are a cycle; The first is arranged according to what is happening; or some presumption; Specialized in people.
Here ibn 'Attiyah mentioned the two options quoted by other exegetes for a specific month:
- Some reports said it was Rajab and as one may read in the above explanation the argumentation for that seems strong,
- while others reported it to be dhul-Qia'dah and both were mentioned by at-Tabari in his explanation of verse (5:2) (see the reports: #10947 for dhil Qi'adah here) and he referred to his opinion made in his explanation of verse (2:217) were he mentioned a lot of incidences and report suggesting that Rajab is the stronger opinion (see here).
