I don't know about a source, but Conway had been compiling knot tables since high school, and as a knot with only 11 crossings, the Conway knot is certain to have been on his radar. It's just such a simple knot, and sliceness such a fundamental property that Conway is sure to have tried to figure it out.
You've caught me there. I'm having trouble even locating a source for why it's even called the Conway knot! The wolfram page has a few references of about the right age, but I don't have access to them, sadly. It being a long-standing question about a knot named after him, I'm fairly certain that he'd take a crack at it. But alas, that's no proof.