#NO.
NO.
Jon was burned by the fire he used to kill the wights in Season 1 that invaded Castle Black.
Quotes from the book, which since this is Season 1, the scenes in the show remain highly faithful to. The burns were quite severe and took weeks or months to heal properly.
Jon plunged his hand into the flames, grabbed a fistful of the burning drapes, and whipped them at the dead man. Let it burn, he prayed as the cloth smothered the corpse, gods, please, please, let it burn.
A Game of Thrones - Jon VII
...
"You do not look well. How is your hand?"
"Healing." Jon flexed his bandaged fingers to show him. He had burned himself more badly than he knew throwing the flaming drapes, and his right hand was swathed in silk halfway to the elbow. At the time he'd felt nothing; the agony had come after. His cracked red skin oozed fluid, and fearsome blood blisters rose between his fingers, big as roaches. "The maester says I'll have scars, but otherwise the hand should be as good as it was before."
A Game of Thrones - Jon VIII
...
He flexed his burned hand as he rode, opening and closing the scarred fingers. They still pained him, but it felt good to have the wrappings off.
A Game of Thrones - Jon IX
...
He gave Jon a long, searching look. "But we were speaking of hands. How is yours?"
"Better." Jon peeled off his moleskin glove and showed him. Scars covered his arm halfway to the elbow, and the mottled pink flesh still felt tight and tender, but it was healing. "It itches, though. Maester Aemon says that's good. He gave me a salve to take with me when we ride."
A Clash of Kings - Jon I
