What is the origin of Naam Traya Mantra? Or in which scripture this mantra is mentioned? The mantra is Om Achutaya namaha Anantaya namaha Govindaya Namaha. I don't know the correct form but these 3 names of God Vishnu are there in Naam Traya Mantra. I also got the Rishi and Chanda from internet but don't know in which scripture this information is found?
These Mantras are found in the Padma Purana and are also used during Achamana by certain sects.
Quoting from the book "Essence of Padma Purana" @Kamakoti.org:
The arrangement of the ‘Samudra Mathana’was that Manthara Mountain was to be set as the churning rod, Maha Sarpa Vasuki was the string with which to tie up the Mountain and Bhagavan himself as balancing Shakti as the Avatar of Kurma (Tortoise) to ensure that the Meru Mountain would be made stable and steady. As the churning started, innumerable of the proud Danavas who opted for the head of thousand faced Vasuki out of pride got perished due to the poisonous flames where as Devas were at the tail end. Eventally, a massive ‘Halahala’ or the sky-rocketing poisonous flames engulfed the Univerese and by the grace of Three Names of Lord Vishnu viz. Om Achyutaya Namah, Om Anantaya Namah and Om Govindaya Namah, Bhagavan Shankara gulped the blue poisonous flames in his throat thus giving him the epithet of Neela Kantha. By the power of the three names of Vishnu, there would never be any hardship faced:
Achyutaananta Govinda iti naamatrayam Maha Mantram hareh/
yo japetpriyato Bhaktaayaa Pranavaadhyam namontakam/
Tasya Mrityu bhayam naasti visharogaagnijam mahat/
Naamatrayam Maha Mantramjapedyam prayatatmavaan/
Kaala Mrityu bhayam chaapi tasya naasti kimanyatah
Whoever recites the Three Names viz. Om Achyutaaya, Om Anantaaya, and Om Govindaaya sincerely and with dedication, none of the poisonous diseases or Agni- based frights/deaths would affect.
Here, as one can see, the context is Samudra Manthana.
But it is also quite possible that the Mantras are also found in some other scriptures as well.
I have found the Rishi-Devata etc for the Nama Traya Mantra in Nityotsava's 3rd Chapter, the Shri Krama. But the they are not matching with the ones you have given. I am giving below the limbs as well as the Dhyanam for the Mantra:
NAmatraya mantrasya kAshyapAtribharadvAjA rishayah | Anushtup Chandah | ShrimahAvishnur devatA | Tat prityarthe jape viniyogah ||
Samastadurastar vyAdhi sangha dhvamsa patiyase |
Achyut ananta govinda nAmne dhAmne namo namaha ||
Note that there are three Rishis - Kasyapa, Atri and BharadvAja - mentioned as the seers of the Mantra.
As per PADMA-PURANA: UTTARAKHANDA: Chapter 78 The Hymn called 'Apamarjana',
1-2. I shall hereafter tell you about the excellent vow of the sprinkling with water, as told by Pulastya to the magnanimous Dalabhya. It destroys all faults caused by diseases. It gives auspiciousness. I shall tell it to you. Listen, O daughter of the (Himalaya) Mountain.
Sri Pulastya said [to Sri Dalabhya]:
45-65. ... By means of the medicine of the utterance of the names, viz. Acyuta, Ananta, Govinda, all the diseases vanish. ...
It's references can be found during Samudra Manthan episode also.
PADMA-PURANA: BHUMIKHANDA: Chapter 9. The Churning Starts:
2-7a. Seeing them running away Shankara said these (words): "O hosts of gods, hand over the poison to me. I shall quickly inactivate the great (i.e. severe) poison, viz. Kalakuta". > Saying so, that lord of Parvati, meditating upon Visnu in his heart and uttering the great hymn, took that fearful poison. By the power of the great hymn, that great (deadly) poison was digested.
7b-22. 7b-22. He who, being controlled, would devoutly mutter the three names, viz. Acyuta, Ananta, and Govinda, of Visnu with Pranava (i.e. Om) at the beginning and Om at the end, has no fear of taking poison, or from fire and also from death. ...
PADMA-PURANA: UTTARKHANDA: Chapter 232. The Raise of Goddess Laksmi:
16-21. Having meditated (i.e. when I meditated) with a concentrated mind upon that lord along with J§rl and Bhumi, having earrings of gold purified by fire, remover of all miseries, and on my having muttered the great hymn of the name and form (of Visnu) along with Mahalaksmi, all that poison, which was very fierce, which was the first one, which was fearful to all, which was (capable of) destroying all the worlds, was digested (by me) due to the three names of Visnu, the omnipresent one. He who, being restrained, would devoutly mutter the three names of Visnu, viz. Acyuta, Ananta, Govinda, beginning with Pranava (i.e. Om) and (also) ending with Om, has no fear of death, so also the great fear due to poison, disease and fire. The wise, restrained one, who would mutter the great hymn—the three names—does not have fear of Death; then from where else (can he have it)? Thus with (the muttering of) the three names I drank that poison.