I don't recommend using carbon during a cycle, but it is possible. The carbon will absorb ammonia
in fact, carbon only adsorbs Organic compounds. ammonia (NH3) is an Inorganic compound and as such, is NOT adsorbed by carbon.
personally, i don't recommend the use carbon in a filter at all and particularly while cycling.
carbon beds typically last only a month at best and more typically 2-3 weeks since most hobbyists tanks are overstocked. at some point, you're going to want to replace that carbon and when you do, it will have already developed a significant concentration of those bacteria you've worked so hard to cycle your tank. thus when you replace it, you'll be tossing out all those bacteria and will likely effect a breakdown in the cycle and the start of a second cycle.
moreover, there are many types of carbon sold for aquarium use but very few of these are actually capable of adsorption in water. only high quality activated granular carbon which has been degassed in an oxygen oven at temperatures close to 2,000 degrees Fahrenheit is functional --- anything else is pretty much worthless. degassing enlarges the apparent surface area of each carbon granule, thus greatly increasing the adsorptive capacity.
a great many carbon grades are washed with phosphoric acid in an effort to support the adsorption sites since this is cheaper than degassing. this phosphate will leach into your aquarium water and can reult in a major algae problem.
deep beds of Granular Activated Carbon (GAC) are always more effective than shallow ones of equal area. also, for a given amount of GAC the smaller the granules, the greater the adsorptive capacity.
generally, there needs to be 4 to 10 grams of GAC for every gallon of water depending on stocking density. if your filter uses prepackaged cartridges containing carbon, realize that these cartridges typically contain poor quality carbon and if quality carbon is used, only 20 percent or less of the recommended amounts of GAC is present. you can slit open the cartridge and fill them with more GAC but many pad filter types simply will not hold that much.
so, unless you're removing medications from your water, or other dissolved organic compounds which would include colors (tannins) and odors, you can really save your money and not use carbon in your filter. regular 50% water changes and gravel vacuuming are as effective except in extreme cases.