Short answer
Yes. The prophet the Quran calls Yahya is the one the Bible calls John the Baptist: son of the elderly Zechariah (Zakariya) and a close relative of Jesus. The Quran honours him as a pure and devout prophet who confirmed a word from Allah, which most commentators explain as Jesus.
Yahya is one of the prophets named in the Quran, and believing in him is part of believing in the prophets, as all Muslims agree. Muslims take his story from the Quran and authentic hadith, and do not build belief on details found only in other sources.
The four schools agree on this.
Explanation
Yahya is the Arabic form of the name; English Bibles call him John. The Quran tells how Zakariya, old and with a wife who could not have children, prayed for a righteous heir, and the angels gave him the good news of a son named Yahya, a name, Allah says, given to no one before him.
The Quran describes him beautifully. He was given wisdom and judgement as a boy, tenderness and purity, he was dutiful to his parents and never arrogant, and Allah sends peace on him the day he was born, the day he dies and the day he is raised alive. He was chaste and a prophet among the righteous, and he came confirming a word from Allah. Most commentators, following Ibn Abbas, explain that word as Jesus, whom the Quran calls a word from Allah.
In the Bible, John preaches repentance in the wilderness, baptises people in the Jordan, including Jesus, and is later put to death by the ruler Herod. The Quran does not mention baptism or the details of his death. Islamic tradition does hold that he was killed unjustly, and a shrine linked to him stands inside the Umayyad Mosque of Damascus.
The Prophet ﷺ met Yahya and Jesus together in the second heaven on the night of the Mi'raj, and described them as maternal cousins. Read his full story on Prophet Yahya, with Prophet Zakariya and Prophet Isa, and see is there baptism in Islam.
The evidence
Quran, Surah Maryam 19:7
يَٰزَكَرِيَّآ إِنَّا نُبَشِّرُكَ بِغُلَٰمٍ ٱسْمُهُۥ يَحْيَىٰ لَمْ نَجْعَل لَّهُۥ مِن قَبْلُ سَمِيًّۭا ﴿٧﴾
[He was told], "O Zechariah, indeed We give you good tidings of a boy whose name will be John. We have not assigned to any before [this] name."
The Quran's name Yahya is rendered John in English.
Quran, Surah Aal-i-Imraan 3:39
فَنَادَتْهُ ٱلْمَلَٰٓئِكَةُ وَهُوَ قَآئِمٌۭ يُصَلِّى فِى ٱلْمِحْرَابِ أَنَّ ٱللَّهَ يُبَشِّرُكَ بِيَحْيَىٰ مُصَدِّقًۢا بِكَلِمَةٍۢ مِّنَ ٱللَّهِ وَسَيِّدًۭا وَحَصُورًۭا وَنَبِيًّۭا مِّنَ ٱلصَّٰلِحِينَ ﴿٣٩﴾
So the angels called him while he was standing in prayer in the chamber, "Indeed, Allah gives you good tidings of John, confirming a word from Allah and [who will be] honorable, abstaining [from women], and a prophet from among the righteous."
Yahya came confirming Jesus, the word from Allah.
Quran, Surah Maryam 19:12-15
يَٰيَحْيَىٰ خُذِ ٱلْكِتَٰبَ بِقُوَّةٍۢ ۖ وَءَاتَيْنَٰهُ ٱلْحُكْمَ صَبِيًّۭا ﴿١٢﴾ وَحَنَانًۭا مِّن لَّدُنَّا وَزَكَوٰةًۭ ۖ وَكَانَ تَقِيًّۭا ﴿١٣﴾ وَبَرًّۢا بِوَٰلِدَيْهِ وَلَمْ يَكُن جَبَّارًا عَصِيًّۭا ﴿١٤﴾ وَسَلَٰمٌ عَلَيْهِ يَوْمَ وُلِدَ وَيَوْمَ يَمُوتُ وَيَوْمَ يُبْعَثُ حَيًّۭا ﴿١٥﴾
[Allah] said, "O John, take the Scripture with determination." And We gave him judgement [while yet] a boy And affection from Us and purity, and he was fearing of Allah And dutiful to his parents, and he was not a disobedient tyrant. And peace be upon him the day he was born and the day he dies and the day he is raised alive.
The Quran's description of his character and prophethood.
Sahih al-Bukhari 3887
Sahih... فإذا يحيى وعيسى وهما ابنا الخالة ...
"There were Yahya and Isa, who are cousins through their mothers."
The Prophet ﷺ met them together on the Mi'raj and named their family tie.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.