The Quran and authentic hadith do not mention Samson. So we can't make a decisive claim on the authenticity of his story in the Bible.
The twelve leaders of Israel mentioned in [5:12] could potentially allude to the Biblical judges, but not much authoritative detail is present in the Islamic sources.
Samson ( شمسون ) is briefly mentioned in some Islamic traditions, possibly deriving from Isra'iliyyat:
وقال ابن مسعود : إن النبي - صلى الله عليه وسلم - ذكر رجلا من بني إسرائيل لبس السلاح في سبيل الله ألف شهر ; فعجب المسلمون من ذلك ; فنزلت إنا أنزلناه الآية . خير من ألف شهر ، التي لبس فيها الرجل سلاحه في سبيل الله . ونحوه عن ابن عباس .
Ibn Masud said: The Prophet told us of a person from the Bani Israel who who fought Jihad in the way of Allah for a thousand months. When the people were worried by it the verse: "The Night of Decree is better than a thousand months.
[97:3]" was revealed, and this night is better than the thousand months in which he waged Jihad. Similar to this is narrated from Ibn Abbas.
وهب بن منبه : إن ذلك الرجل كان مسلما ، وإن أمه جعلته نذرا لله ، وكان من قرية قوم يعبدون الأصنام ، وكان سكن قريبا منها ; فجعل يغزوهم وحده ، ويقتل ويسبي ويجاهد ، وكان لا يلقاهم إلا بلحيي بعير ، وكان إذا قاتلهم وقاتلوه وعطش ، انفجر له من اللحيين ماء عذب ، فيشرب منه ، وكان قد أعطي قوة في البطش ، لا يوجعه حديد ولا غيره : وكان اسمه شمسون
Wahb ibn Munabbih said: He was a Muslim. When he was born his mother vowed this to Allah. They lived near a people who were idolators. He waged war alone ... with the jaw-bones of a camel, when he thirsted water came out from between the jaws, he was given strength and iron didn't hurt him. His name was Shamson.
— Tafsir Qurtubi
Tabari also mentions Samson in his Tarikh here and here, with details similar to the Biblical narrative, though the chronology may be messed up.