Suppose we have a contract with payoff $P_Y$ in currency $Y$, where $P_Y$ on a variable in currency $Y$.
To calculate the value in $X$, we take the expected payout under $Y$-numeraire $E_Y(P_Y)$, discount using the discount rates for $Y$, then convert into $X$ by using the current spot rate. No quanto adjustment needed in this case.
Now suppose $P_Y$ is paid in currency $X$ instead, using the spot rate at expiry. This requires a quanto adjustment due to the correlation between the payoff and the FX forward rates.
These give different prices, because there's no quanto adjustment in the first case, while there is the second.
To me, this is counterintuitive. All that's needed to change from the first to the second is just a spot transaction at expiry. Why should they not give equal prices, and why would a quanto adjustment not be needed in the first case?
EDIT: Should be spot rate at inception, not expiry