This is one reason why it's recommended that tires always be mounted with the label at the valve hole on the wheel - so you can find where the puncture was.
If the tire is still half-mounted on the wheel, or you did mount the tire with the label at the valve stem, you can easily find the only two places on the tire that correspond to the location of the puncture in the tube. It's usually accurate enough to use estimates such as "the puncture in the tube was about 1/3 of the way around from the valve stem, so the puncture in the tire has to be about HERE or HERE."
If that's not accurate enough to locate the puncture area, put the tube back in the tire, determine one approximate location, look there, and if you don't find it, flip the tube around and try again.
If that doesn't work, or if you didn't mount the tire or keep track of the tube orientation well enough to do that, you'll have to examine the entire tire for the puncture.
One quick way that's likely to find any foreign object poking through the tire is to run something over the inner surface of the tire and examine any area where something catches. Use something like a cotton ball instead of your fingertips so the location will be marked with a few cotton strands instead of skin and blood.
If you still can't find it, it's time to start a close examination of the entire tire's inner and outer surface. Flex the tire when you're examining any area so you can see if any cut or hole opens up.
Also - don't forget to check your wheel in the locations corresponding to the puncture in the tube.