The N.O and N.C. contacts are normally physically separated by an air-gap with the movable common contact in between. To have continuity between a pair of N.O. and N.C. contacts requires that both the N.O. contact and the N.C. contact be "welded" to the common contact across the air gap! That condition is likely to be fairly obvious upon inspection of the relay, and there is probably no way to "fix" it. However, I suspect you are not measuring what you think you are measuring. Try measuring again with another relay of the same type and report back here with the results. Also, a model number and a pin-out of the suspect relay would help us identify what appears to be wrong. Show us these and explain which pins you are using to measure continuity between.