I had earlier killed some spiders before getting the quest. How can now I get the spider egg if I had no spiders to kill ?

don't worry. Spiders are everywhere. They are one of the wandering monsters in the wilderness, so just wander around for a while.

One spider spawns on bottom of Windmill, one in Gypsy house by lake, spiders in Serpent castle mostly behind 6 closed doors you have to open by coins and levers.

I had wandered for many hours and yet no spiders. Any other ways ? I had read elsewhere that the quest was not after a spider egg but a raptor egg at the snake river. Is this true ?

Oh, you doing Ranger quest? You need raptor egg then. After you start ranger quest, raptor spawns in river area, you kill it and it drops raptor egg for ranger quest. Looks really raptorish and it's tough.