During the conference, John Whitmer "gave a short relation of the facts connected with the translation of the book of Mormon." No further details were given in the report, but the reporter (presumably Warren Cowdery) made this observation:
On reflecting how many foolish reports are in circulation on this subject, and how many there are who are vain enough to believe them, I could not but wish that such were present, while Elder [John] Whitmer was delivering his address.
We can all wish we were present when John Whitmer delivered his address. Presumably he corroborated what Oliver Cowdery said about the translation, which Joseph Smith himself also later corroborated multiple times. Oliver was the presiding officer at the meeting and spoke before and after John Whitmer did.
In the context of the times, the "foolish reports" undoubtedly referred to those published in October 1834 in Mormonism Unvailed, which related the stone-in-the-hat theory (SITH) and a garbled version of the Urim and Thummim account that, like SITH, portrayed Joseph as not even referring to the plates. Mormonism Unvailed added the observation that any testimony from witnesses of the plates was pointless if Joseph didn't use the plates.
The gist of Mormonism Unvailed was speculating what was behind the "vail" when Joseph dictated the text. It was common knowledge that Joseph had not shown the plates or U&T during the translation. He dictated from behind a curtain or screen. Mormonism Unvailed proposed that Joseph was reading from a manuscript originally written by Solomon Spalding.
The book set up a Catch 22 problem. If Joseph was dictating from behind a "vail" or curtain, it was anyone's guess what he was reading from. But if Joseph was dictating within the view of others by reading from a stone (or the U&T) that he put in a hat, then he wasn't actually translating the plates and the plates were not evidence of what he was dictating.
To counter the arguments in Mormonism Unvailed, Oliver Cowdery published his unambiguous account the same month Mormonism Unvailed was published.
These were days never to be forgotten—to sit under the sound of a voice dictated by the inspiration of heaven, awakened the utmost gratitude of this bosom! Day after day I continued, uninterrupted, to write from his mouth, as he translated with the Urim and Thummim, or, as the Nephites would have said, ‘Interpreters,’ the history or record called ‘The Book of Mormon.’
(Joseph Smith—History, Note, 1)
When read in historical context, Oliver's account expressly repudiated both prongs of the Mormonism Unvailed Catch 22. While Joseph did dictate the text from behind a "vail" (and did not merely read words off a stone in a hat), he was using both the plates and the Urim and Thummim; i.e., he did not read the Solomon Spalding manuscript.
Although the report of the New Portage conference did not detail what John Whitmer said--did not itemize his "short relation of the fact connected with the translation of the book of Mormon" there--John was one of the Eight Witnesses and, in later years, John testified that Joseph used the U&T and breastplate while translating the plates of Nephi in Fayette.
Zenas Gurley reported that John Whitmer told him “When the work of translation was going on he [John] sat at one table with his writing material and Joseph at another with the breast-plate and Urim and Thummim. The latter were attached to the breast-plate and were two crystals or glasses, into which he looked and saw the words of the book. The words remained in sight till correctly written, and mistakes of the scribe in spelling the names were corrected by the seer without diverting his gaze from the Urim and Thummim.” In S. F. Walker, "Synopsis of a Discourse by Zenas H. Gurley," The Saints Herald (December 15, 1879, vol 26, issue 24), pp. 369 – 371, at p. 370, available at https://archive.org/details/TheSaintsHerald_Volume_26_1879/page/n369/mode/2up .
Here is the report of the conference that was published in the Messenger and Advocate, June 1835, Vol. 1, 9:142, available on WordCruncher and at this website: http://www.latterdaytruth.org/pdf/100110.pdf
Excerpts from the report:
NEW PORTAGE CONFERENCE.
Oliver Cowdery “New Portage Conference,” M&A 1 (Jun 1835)
This meeting of the elders and brethren was a joyous one—the number of elders I do not recollect, but there were some eighteen or twenty, representing several little branches of the church. The brethren from a distance were in good spirits, and manifested an unshaken confidence in the gospel which they had embraced. The church at New Portage numbers one hundred and more, many, or the most of whom, were present at the meeting on the Sabbath. ...
On Saturday the 6th, the elders assembled in conference, in a large and convenient room, furnished by elder A. Palmer, the presiding elder of that church. Elder O. [Oliver] Cowdery was unanimously called to preside, and elder W. [Warren] A. COWDERY, from Freedom, N. Y. chosen Secretary. ...
After an able and fervent address to the throne of grace, elder O. [Oliver] Cowdery delivered an interesting discourse upon the plan and order of heaven in the salvation of the human family, followed by elders P. [Phineas] H. Young, Z. [Zebedee] Coltrin, & A. [Andrew] J. Squiers; after which Elder [Ambrose] Palmer gave an invitation to those who desired, to be baptized, when three came forward and were buried in the liquid grave. This was an interesting season, and many of the by- standers were, apparently, touched with a sense of the importance of that moment when an individual steps forward, in the presence of this world and the heavenly hosts, and covenants to follow the Lamb of God who takes away his sins.
Elder John Whitmer took the lead in the services of the afternoon, and gave a short relation of the facts connected with the translation of the book of Mormon. On reflecting how many foolish reports are in circulation on this subject, and how many there are who are vain enough to believe them, I could not but wish that such were present, while Elder [John] Whitmer was delivering his address.
A thousand things may be conjectured, but when a man declares openly, candidly, and seriously, of what he has seen, hefted and handled with his own hands, and that in the presence of a God who sees and knows the secrets of the heart, no man possessed of common reason and common sense, can doubt, or will be so vain as to dispute.
Such is the fact that a record of that description does exist, for it has been seen, and such is the fact, that the Lord himself bears witness of it, for thousands testify of the same—there is neither lack of human or divine testimony: Then who so blind as not to see? And who so deaf as not to hear?
Elder [John] Whitmer was followed by several elders, and the meeting closed with a few remarks from elder O. [Oliver] Cowdery upon the further truth of the book of Mormon. The meeting was continued till quite late, after which one more came forward and was baptized.
(Messenger and Advocate I.9:142-3 )