Here's a natural picture which suggests not: genuine disagreement requires a dispute over the (objective, "God's eye view") state of the world. This explains why, if tomorrow I say "It's raining", I don't really disagree with your present claim that it isn't raining. Both can be right, because there's no objective matter of fact about which we disagree. (We can both agree that it rains on March 19, 2009, but not March 18.) The situation is similar, we may think, for the cultural relativists who disagree about whether abortion is wrong. Both may agree that it is condoned by Sally's moral standards but not by Anne's, and at that point it is difficult to see what is left for them to disagree about. (They might fight, or have conflicting desires, but that is not the same thing as disagreeing in their propositional beliefs. It's more like the dinnertime 'disagreement' between a predator and its prey.)
The relativist might insist that they can still disagree over the relativistic proposition whether abortion is wrong. But sophisticated interlocutors will recognize that this effectively amounts to disagreement over whether abortion is wrong relative to the assessor's moral standards, which again seems to reduce to a non-cognitive conflict between the two moral standards. (It's not as though either party is making any kind of rational/epistemic error -- a fact, incidentally, that should give the moralist pause.)
So it's unsatisfying for the relativist to just appeal to formalities, e.g. "we disagree over the truth of this proposition!" That seems too cheap. But perhaps they can say more. On p.19 of 'Relativism and Disagreement' [pdf], MacFarlane suggests:
Accuracy [i.e. truth at the relevant context of assessment] is the property we must show assertions to have in order to vindicate them in the face of challenges, and it is the property we must show others’ assertions not to have if our challenges are to be justified.
This suggests the following account of genuine disagreement: two people genuinely disagree if they can appropriately challenge each other's assertions.
Sometimes people challenge assertions inappropriately. For example, if tomorrow I were to look back and say, "You said it wasn't raining, but now look: it clearly is!" my challenge would be inappropriate. Even given my beliefs, your past assertion was perfectly accurate, since the relevant context of assessment, in this case, is the context of utterance -- a time at which (I know) the statement was indeed true. So this account suffices to explain standard cases of merely apparent disagreement. But it does so in a way which is (at least potentially) compatible with maintaining genuine disagreement in case of relative truths.
The big question, now, concerns what is the "relevant" context of assessment when evaluating assertions about a relativistic domain. Again, suppose moral relativism is true. When we assess Anne's claim that abortion is wrong, is it Anne's moral standards or our (the assessor's) own that are relevant? To preserve moral disagreement, the relativist will want to insist on the latter. I argue elsewhere that this is a mistake: the relevant context for moral assessment is always the actor's. But, in any case, this is the crux of the dispute, and I'm more sympathetic to MacFarlane's line in other domains, e.g. matters of taste, humour, etc.
P.S. It seems to me that there's something to be said for both criteria of genuine disagreement discussed above. I think MacFarlane's is plausibly the primary notion: philosophical inquiry into disagreement is initially motivated by concern about our assertoric practices, which is what he focuses on. Yet there seems an important sense in which (from a philosopher's rationalistic perspective) relativistic disagreement is defective. So even if we grant that MacFarlane has correctly diagnosed what constitutes "genuine disagreement", we might still want to reserve a related term -- "objective" or "substantive disagreement", say -- for disagreements that meet the first criterion, of disputing how things are objectively.