In the book A Game of Thrones the great-sword known as Ice is not mentioned in the encounter between Jaime and Ned.
I have quoted select sentences from the full passage that details the fight.
"No!" Ned Stark screamed, clawing for his sword. Jaime was already cantering off down the street as he heard Wyl shout.
Ned galloped to him, bringing his longsword down on Tregar's helm.
Ned's sword is only mentioned twice and is never called "Ice" or even a "great-sword". Ice is always called by name or as a "great-sword", if not both in the books.
Ned was also mounted on horse in the books. It would be very difficult to wield a two-handed great-sword while mounted, so even if Ned had Ice on his person at the time of the fight, he would have wanted to use something more suitable for the situation. The only time we see Ice used is for executions.
Remember Valyrian steel swords are very rare and treasured. Not even the Lannisters have enough gold to make a lesser house forget their pride and honor to sell one.