The forward rate that applies to a interval of time ending at T is a martingale under Q^T, the measure that has P(·, T) as the numeraire, thus no adjustment is required. But the same forward rate is not a martingale under Q^(T-1), so a convexity adjustment is needed. This is used, for example, when pricing in-Arrears swaps. The file www.cam.wits.ac.za/mfinance/projects/nevena.pdf
is quite clear and explains very well this issue.