Probably not
Outside of OWL and NEWT years (fifth and seventh, respectively), the end of year exams are conducted by the regular teachers, presumably during the normally scheduled period for the class. We know that lessons are split by House, or taken at the same time by two Houses, but neither Transfiguration or Charms is a "shared" subject - Gryffindor attends lessons for both classes with only their House. Therefore, Hermione wouldn't need to be the only third year student taking Arithmancy and Ancient Runes to be the only one with a conflict, she would just need to be the only Gryffindor.
Unfortunately there's no explicit proof in the books that she's not the only student, unlike for Muggle Studies:
"... And Ernie McMillan told me she's never missed a Muggle Studies class, but half of them are at the same time as Divination, and she's never missed one of them either!"
Harry Potter and the Prisoner of Azkaban, Chapter Twelve - The Patronus
However, it wouldn't make sense for Hogwarts to run the class for a single student, especially if that student wanted to take on such a heavy workload that she required a Time-Turner.
In terms of being the only Gryffindor, all other Gryffindor third years are shown in both Divination and Care of Magical Creatures lessons.
Divination
Neville:
"... You, boy," she said suddenly to Neville, who almost toppled off his pouf. "Is your grandmother well?"
Harry Potter and the Prisoner of Azkaban, Chapter Six - Talons and Tea Leaves
Parvati:
"... By the way, my dear," she shot suddenly at Parvati Patil, "beware a red-haired man."
ibid.
Lavender:
"I wonder, dear," she said to Lavender Brown,who was nearest and shrank back in her chair, "if you could pass me the largest silver teapot?"
ibid.
Dean:
"What is it, Professor?" said Dean Thomas at once.
ibid.
Seamus:
Seamus Finnigan was tilting his head from side to side.
ibid.
Care of Magical Creatures
Lavender:
"Oooooooh!" squealed Lavender Brown, pointing toward the opposite side of the paddock.
ibid.
Parvati:
There was an intake of breath from behind him, and both Lavender and Parvati whispered, "Oooh, no, Harry, remember your tea leaves!"
ibid.
Neville:
Neville ran repeatedly backward from his, which didn't seem to want to bend its knees.
ibid.
Dean:
"It was Malfoy's fault!" snapped Dean Thomas.
ibid.
Seamus:
"First pus and now this," muttered Seamus.
Harry Potter and the Goblet of Fire, Chapter Thirteen - Mad-Eye Moody