Here are the narrations which prove that the birth of the Imams (as) is different from that of normal Human beings:

Hadith 1:

Hakima bint Mohammad Taqi (ع) when in the house of Imam Hasan Askari (ع) on the night of the arrival of the Imam al-Qaim (ع) says;

فقلت لسيدي أبي محمد عليه السلام: ما أرى بها حملا فتبسم عليه السلام ثم قال: إنا معاشر الاوصياء لسنا نحمل في البطون وإنما نحمل في الجنوب ولا نخرج من الارحام وإنما نخرج من الفخذ الايمن من امهاتنا لاننا نور الله لا تناله الدانسات

I said to my Master Imam Hasan Askari (as), "I do not see any pregnancy in Her." Imam (as) smiled and said: "We successors (Awsiya) are not carried in the abdomens (belly), but rather, we are carried on the sides. We do not come out from wombs, but rather from the right leg of Our mothers. Because we are the light (Noor) of Allah to which impurities do not reach."

[Source: Bihar al-Anwar, Vol. 51, Pg. 26]

Hadith 2:

In a narration about the birth of Imam al-Qaim (عليه السلام), Imam Hasan Askari (عليه السلام) says,

قال : إنَّ أولاد الأنبياء والاصياء إذا كانوا أئمّة ينشؤون بخلاف ما ينشؤ غيرهم ، وإنَّ الصبيَّ منّا إذا كان أتى عليه شهرٌ كان كمن أتى عليه سنّة ، وإنَّ الصبي منا ليتكلّم في بطن اُمّه ويقرأ القرآن ويعبد ربّه عزَّ وجلَّ ، [ و ] عند الرِّضاع تطيعه الملائكة وتنزل عليه صباحاً ومساءً

"Indeed, the children of the Prophets and the successors (of the prophets), if they are Imams (leaders), grow up differently from how others grow up. And indeed, a child among us, if a month has passed upon him, is like one upon whom a year has passed. And indeed, a child among us speaks in his mother's womb and recites the Quran and worships his Lord (عز وجل). And at the age of nursing, the angels obey him and descend upon him morning and evening." 

[Source: Kamal al-Deen, Shaikh Saduq, Chapter. 42, Hadith. 2]

Hadith 3:

وروي عن آمنة أم النبي صلى الله عليه وآله انها قالت لما حملت برسول الله صلى الله عليه وآله لم اشعر بالحمل ولم يصبني ما يصيب النساء من ثقل الحمل ورأيت في نومي كأن آتيا اتاني فقال لي قد حملت بخير الأنام ثم وضعته يتقي (قابض ك) الأرض بيديه وركبتيه ورفع رأسه إلى السماء وخرج مني نور أضاء ما بين السماء إلى الأرض 

It is narrated from Amina (ع) the mother of the Prophet (ص) that She said, "When I carried (pregnant with) the Prophet (ص), I did not feel the pregnancy and I was not affected with that which affects women due to the weight of the pregnancy. And I saw in my dream as a person coming, He came to Me and said to Me, "Verily You are carrying the best of the creations, then He (Prophet Muhammad ص) came out and placed His hands and knees on the ground and raised his head towards the skies and emerged from me a light illuminating what is between the skies and the earth."

[Source: Tafseer al-Qummi, Vol. 1, Pg. 373]

Hadith 4:

إكمال الدين: بهذا الاسناد عن محمد بن عثمان العمري قدس الله روحه أنه قال: ولد السيد (ع) مختونا وسمعت حكيمة تقول: لم ير بامه دم في نفاسها وهذا سبيل امهات الائمة صلوات الله عليهم

Through the same chain of narration from Muhammad Ibn Othmān al-Amari who said,

"The Master was born circumcised and I heard Lady Hakima say, 'No blood was seen in her mother at his birth.' This is the manner of the mothers of the Imams (ع)."

[Source: Kamal al-Deen, Sheikh Sadooq, Chapter. 42, Hadees. 14, Pg. 433]

Hadith 5:

In a narration detailing the arrival of the 12th Imam (ع), Hakima bint Imam Mohammad Taqi (ع) says about the mother of al-Qaim (ع), Hazrat Nargis (ع):

و لم أر بها حملا و لا أثر حمل

"And I did not see in Her any pregnancy & no signs of pregnancy"

[Source: Dalaail al Imamah, Pg 497]

Hadith 6:

حَدَّثَني تَمِيمُ بْنُ عَبْدِاللَّه بْنِ تَمِيمٍ القُرَشِي رَضِىَ اللهُ عَنْهُ قالَ: حَدَّثَني أَبي، عَنْ أَحْمَدَ بْنِ عَلِى الأَنْصارِي، عَنْ عَلِىِّ بْنِ مِيْثَم، عَنْ أَبيهِ قالَ: سَمِعْتُ اُمِّي تَقُولُ: سَمِعْتُ نَجْمَة اُمِّ الرِّضا عَلَيْهِ السَّلامُ تَقُول: لَمّا حَمَلْتُ بِاِبْنِي عَلِيٍّ لَمْ أَشعِرْ بِثِقْلِ الحَمْلِ، وَكُنْتُ أَسْمَعُ فِي مَنامِي تَسْبِيحاً وَتَهْلِيلاً وَتَمْجِيداً مِنْ بَطْني، فَيَفْزَعُنِي ذلِكَ وَيَهُولُنِي فَإِذا انتَبَهْتُ لَمْ أَسْمَعُ شَيْئاً، فَلَمَّا وَضَعْتُهُ وَقَعَ عَلَى الأَرْض واضِعاً يَدَيْهِ عَلَى الأَرْض، رافِعاً رَأسِهِ إِلى السَّماءِ، يُحَرِّكُ شَفْتَيْهِ كَأَنَّهُ يُتَكَلِّمُ فَدَخَلَ إِلىَّ أَبُوه مُوسَى بْنُ جَعْفَرٍعَلَيْهِمَا السَّلاَمُ فَقالَ لي‏: هَنِيئاً لَكِ يا نَجْمَةُ كَرامَةُ رَبِّكَ فَناوَلْتُهُ إِيَّاهُ فِي خِرْقَةٍ بَيْضاءَ، فَأَذِّنَ فِي اُذُنِهِ الُيمْنى‏ وَأَقامَ اليُسْرى‏ وَدَعا بِماءِ الفُراتِ فَحَكَنَّهُ بِهِ، ثُمَّ رَدُّهُ إِلَىَّ، فَقالَ: خُذِيهِ، فَإِنَّهُ بَقِيَّةُ اللَّه تَعالى‏ فِي أَرْضِهِ

Tamim ibn Abdullah ibn Tamim al-Qurashi narrated that his father quoted on the authority of Ahmad ibn Ali Al-Ansari, on the authority of Ali ibn Maysam on the authority of his father,

"I heard Najma, the mother of Imam Al-Ridha (ع), say, "When I was pregnant with my son Ali (ع), I did not feel any burden. I used to hear in my sleep Tasbeeh (Praise), Tahleel, and Tamjeed (glorification) from my womb. So that made me scared and then when I woke up I did not hear anything."

"Once I gave birth to my child, he put both of his hands on the ground, raised His head towards the sky, and moved his lips as if he was saying something. His father, Musa ibn Jafar (ع) came to me said, "O Najma! Congratulations on this great honor (Karamah) of Your Lord."

Then I handed Him to Imam al-Kazim (ع) in a white cloth, He (ع) said the Azan in حis right ear and the Iqamah in حis left ear. Then He (ع) asked for some water from the Euphrates River (Furaat) and gave Him some of it. Then he gave the child back to me and said,  "Take him. He is Allah's Continuation (Baqiyyatullah) on the Earth."

[Source: Uyoon al-Akhbar, Shaikh Sadooq, Vol. 1, Ch. 3, Hadith. 2]

Hadith 7:

حدثنا محمد بن علي بن حمزة بن الحسن بن عبيد الله بن العباس بن علي بن أبي طالب عليه السلام، قال: سمعت أبا محمد عليه السلام يقول: قد ولد ولي الله وحجته على عباده وخليفتي من بعدي مختونا ليلة النصف من شعبان سنة خمس وخمسين ومائتين عند طلوع الفجر، وكان أول من غسله رضوان خازن الجنان مع جمع من الملائكة المقربين بماء الكوثر والسلسبيل، ثم غسلته عمتي حكيمة بنت محمد بن علي الرضا عليهم السلام 

Narrated Abu Muhammad Imam Hasan Askari (عليه السلام) said,

"The Wali of Allah and His Proof (Hujjah) upon his servants and My successor after Me, was born circumcised on the night of 15 Shaban in year 255 during the time of Fajr. And the first one who washed (gusl) Him was Ridwan the gate keeper of heaven along with a group of close angels with the water of al-Kausar and al-Salsabeel. Then My aunt Hakimah bin Muhammad bin Ali al-Rida (عليهم السلام) washed Him."

[Source: Mukhtasar Isbaath al-Rajah, al-Fadhl bin Shazaan, Hadith. 11]

Hadith 8:

حدثنا عبد الواحد بن محمد بن عبدوس العطار رضي الله عنه قال: حدثنا علي ابن محمد بن قتيبة النيسابوري، عن حمدان بن سليمان، عن محمد بن الحسين بن [ي‍] زيد، عن أبي أحمد محمد بن زياد الأزدي قال: سمعت أبا الحسن موسى بن جعفر عليهما السلام يقول - لما ولد الرضا عليه السلام -: إن ابني هذا ولد مختونا طاهرا مطهرا، وليس من الأئمة أحد يولد إلا مختونا طاهرا مطهرا، ولكن سنمر الموسى عليه لإصابة السنة واتباع الحنيفية

Narrated that Abu al-Hasan Musa bin Jafar (عليهما السلام) said the following when al-Rida (عليه السلام) was born:

"This son of mine was born circumcised, clean (tahir) and pure (Mutahhar) and none from the Imams is born except circumcised,  clean (tahir) and pure (Mutahhar), but We move the knife over him just in keeping up with the practice (Sunnah of Circumcision) and in following the true religion (al-Hanifiyah)."

[Source: Kamal al-Deen, Vol. 1, Pg. 461]

Hadith 9:

أخبرنا محمد بن محمد، قال: أخبرني أبو القاسم جعفر بن محمد ابن قولويه، عن أبيه، عن سعد بن عبد الله، عن أحمد بن محمد بن عيسى، عن موسى ابن طلحة، عن علي بن أبي حمزة، عن أبي بصير، قال: سمعت أبا عبد الله جعفر بن محمد (عليه السلام) يقول: إن في الليلة التي يولد فيها الإمام لا يولد مولود إلا كان مؤمنا، وإن ولد في أرض الشرك نقله الله إلى الايمان ببركة الامام 

Narrated that Abu Abd-Allah Imam Sadiq (عليه السلام) said:

"Indeed the night in which The Imam (ع) is born, no one is born but that he is a believer. And if he is born in the land/family of unbelievers, Allah will turn him towards faith due to the blessings of the Imam."

[Source: al-Amali, Shaikh Toosi, Pg. 412]