Can you explain what you mean? This definition for meritocracy: "government or the holding of power by people selected on the basis of their ability." says nothing about fairness. If people are able to gain increased ability based on privilege than you have an unfair meritocracy. Since it is pretty clear that you can indeed do this, through parental involvement, tutoring, nutrition, better medical care, etc. than it follows that meritocracy is not fair.

There's lots of definitions, I think this one is probably closer to what we think of when discussing meritocracy:

an elite group of people whose progress is based on ability and talent rather than on class privilege or wealth

http://www.dictionary.com/browse/meritocracy

There is still no conflict though, so long as class privilege and wealth can be transformed into ability and talent.

Well no because the definition is 'rather than class privilege or wealth.' Transforming wealth into ability and talent is class privilege.

If ability comes from wealth, it's still ability. I'd read "rather than class privilege or wealth" to mean that those things aren't considered merits in and of themselves, except to the degree that they can help someone achieve their full potential.

Meritocracy is very much in conflict with equality. Meritocracy means you are judged on your merit. It is completely orthogonal to a nature-vs-nurture argument.

For example, this is exactly the sort of debate we've been having in Singapore, which prides itself on being a highly 'meritocratic', which consequently people have criticized for exacerbating income/social inequality.

Equality vs equity.

Most people intend the latter when they mean the former (at least from usage).